1. RELATION AND FUNCTIONS
Definitions and formula :
1. A relation R in set A is said to be reflexive, if (a, a) Є R V a Є A
2. A relation R in set A is said to be symmetric if (a, b) Є R => (b, a) Є R, a, b Є
A
3. A relation R in set A is said to be transitive if (a, b) Є R & (b, c) Є R => (a, c)
Є R for a, b, c Є A
4. A relation R in set A is said to be equivalence relation if relation R is reflexive,
symmetric and transitive.
5. One – One function : A function f : A  B is said to be one – one (or
injective), if for every a, b Є A, a ≠ b => f (a) ≠ f (b) or we can also say that f
(a) = f (b) => a = b.
6. Onto function : A function f : A  B is said to be onto function (or surjective),
if for every b Є B, there exist an element a Є A such that f (a) = b.
7. Composition of function : Let f : A  B and g : B  C be two given
functions. Then composition of function ‘f’ and ‘g’ is a function from A to C and
is denoted gof. We have (gof)(x) = g (f(x)) V x Є A.
8. A function f : A  B is said to be invertible, if there exists a function g : B  A
such that gof = IA and fog = IB. The function ‘g’ is called the inverse of function
‘f’ and is denoted by f–1.
9. If f : A  B, g : B  C and h : C  D are the given functions, then ho (gof) =
(hog) of .
10. If f : A  B and g : B  C be two invertible functions then gof is also invertible
and (gof)–1 = f–1og–1.
11. Binary operation : A binary operation * on set A is a function * : A x A  A and
is denoted by a * b
12. A binary operation * on set A is said to be commutative if for all a, b Є A, a * b
= b * a.
13. A binary operation * on set A is said to be associative if for all a, b, c Є A,
(a * b) * c = a * (b * c)
14. Given a binary operation * : A x A  A, an element e Є A, if exist is called
identity element for binary operation *, if a * e = e * a = a V a Є A
15. Given a binary operation * : A x A  A with identity element e Є A. An
element a Є A is said to be invertible with respect to the binary operation *, if
there exist an element b Є A such that a * b = e and b * a = e, b Є A is called
inverse of a Є A and is denoted by a–1.
RELATIONS AND FUNCTIONS :
1. Let A = { 1,2,3,4,5} and B = {1,2,3,….,6,7}. If R be a relation from the set A to
the set B defined by (i) is square of (ii) is cube root of , find R and also its
domain and range.
2. Let R be the relation on Z defined by R = {(a, b), a, b Є Z, a 2 = b2}.
Find (i) R
(ii) Domain of R
(iii) Range of R.
3. Determine the domain and range of the following relations on R :
i)
R1 = { (x, 1/x) : 0 < x < 6, x Є N }
ii)
R2 = { (x, x2 + 7) : x is an even natural number }.
4. Let A = {1,2,3,4}. Find which of the following relations on A are reflexive,
symmetric, transitive :
i)
R1 = { (1, 1), (2, 2), (3, 3), (4, 4), (1, 3), (3, 4)}
ii)
R2 = {(2, 4), (4, 2)}
iii)
R4 = A x A
5. Consider the relation ┴ (perpendicular) on a set L of lines in a plane. Show
that this relation is symmetric and neither reflexive nor transitive.
6. Let R be the relation on the set IR of real numbers defined by (a, b) Є R iff (1
+ ab > 0). Show that the relation R is reflexive, symmetric but not transitive.
7. Test whether the following relations are reflexive, symmetric or transitive :
i)
R1 on Z defined by (a, b) Є R1 if
(|a–b| ≤7)
ii)
R2 on Q defined by (a, b) Є R2 if ( ab = 4 )
iii)
R3 on IR defined by (a, b) Є R3 if (a2 – 4ab + 3b2 = 0)
8. Given the relation R = { (1, 2), (1, 1), (2, 3) } on the set A = {1, 2, 3}, add the
minimum number of elements of A x A to R so that the enlarged relation is
reflexive, symmetric and transitive.
9. Let A = {1, 2, 3}. Show that none of the following relations on A is an
equivalence relation :
i)
R1 = { (1, 1), (2, 2), (2, 3), (3, 2)}
ii)
R2 = { (1, 1), (2, 2), (3, 3), (1, 3), (3, 2)}
iii)
R3 = { (1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}
10. Let R be the relation of Congruency on the set A of all triangles in a plane.
Show that the relation R is an equivalence relation.
11. Let integer m be related to another integer n iff m is a multiple of n. Show
that this relation is not an equivalence relation.
12. Let m be a fixed positive integer. Two integers a and b are said to be
congruent moldulo m, written a ≡ b (mod m) if m divides a – b. Show that the
relation of congruent module m is an equivalence relation.
13. Show that the relation R on the set IN x IN defined by (a, b) R (c, d) iff (a + d
= b + c) is an equivalence relation.
14. Show that the relation IR on the set IN x IN defined by (a, a) R (c, d) iff
(ad (b + c) = bc (a + d)) is an equivalence relation.
15. If IR is an equivalence relation on a set A then show that the inverse relation
IR–1 of IR on A is also an equivalence relation.
16. If R and S are equivalence relations on a set A then show that the relation R
∩ S on A is also an equivalence relation.
FUNCTIONS :
1. Let A = {1, 2, 3, 4}, B = {1, 6, 8, 11, 15}. Which of the following are functions
from A to B?
i) f : A  B defined by f(1) = 1, f(2) = 6, f(3) = 8, f(4) = 8.
ii) f : A  B defined by f(1) = 1, f(2) = 6, f(3) = 15, f(3) =
iii) f : A  B defined by f(1) = 6, f(2) = 6, f(3) = 6, f(4) = 6.
iv) f : A  B defined by f(1) = 1, f(2) = 6, f(2) = 8, f(3) = 8, f(4) = 11.
2. If f(x) = x2, find [ f(1.1) – f(1) ] / [ (1.1) – 1 ]
3. The function ‘t’ which maps temperature in celsius into temperature in
Fahrenheit is defined by t(c) = (9C / 5) + 32.
Find (i) t (0) (ii) t (28)
(iii) t (-10)
(iv) The value of C when t(c) =
212.
4. Let f : R  R be defined as
x≤4
f(x) = 2x + 1
x+4
x≥4
Show that f is not a function.
5. If f : R  R is defined by f(x) = x2 – 3x + 2, find f(f(x)). Also evaluate f (f (5)).
6. Find which of the following functions from A to B are one – one when A = {1,
2, 3, 4} and B = {1, 4, 7, 8}.
i)
f : A  B defined as f(1) = 1, f(2) = 4, f(3) = 4, f(4) = 8
ii)
f : A  B defined as f(1) = 4, f(2) = 7, f(3) = 1, f(4) = 8
7. Which of the following functions are one – one ?
i)
f : R  R defined by f(x) = 4,
xЄR
ii)
f : R  R defined by f(x) = 6x – 1,
xЄR
iii)
f : R  R defined by f(x) = x2 + 7,
xЄR
iv)
f : R  R defined by f(x) = x3,
v)
Ф : R – {7}  R, defined by Ф(x) = (2x + 1) / (x – 7 ), x Є R – {7}
xЄR
8. Which of the following functions are onto ?
i)
f : IR  R defined by f(x) = 115x + 49, x Є R
ii)
f : IR  IR defined by f(x) = | x | ,
xЄR
iii)
f : IR  IR+ defined by f(x) = √x ,
xЄR
iv)
f : IR  R+ defined by f(x) = x2 + 4,
xЄR
9. Let f : IR  IR be a function defined by f(x) = cos (2x + 3). Show that this
function is neither one – one nor onto.
10. Iff : IR  IR be a function defined by f(x) = 4x3 – 7. Show that the function f
is a bijective function.
11. A function f : N  Z is defined by
f (n) = (n – 1) / 2
- (n / 2
if n is odd
)
if n is even
Show that this function is a bijection.
12. Let A = [ - 1, 1 ] and f : A  A be a function defined by f(x) = x | x |. Show that
f is a bijection.
13. Let A = {1,2,3} and B = {4,5,6}. f : A  B is a function defined as f(1) = 4,
f(2) = 5, f(3) = 6. Write down f–1 as a set of ordered pairs.
14. Find the inverse of the function f(x) = 4x – 7 , x Є R
15. If f : R – {3 / 5}  R – {3 / 5} be a function defined by
f (x) = (3x + 2) / (5x – 3) , x Є R – {3 / 5}.
Show that f
–1(x)
= f(x) ,
x Є R – {3 / 5}
16. Show that the function f : N  N defined by
f (x) = x2 + x + 1 ,
x Є N is not invertible.
17. Let f : N  N be defined by f(x) = 2x ,
not invertible.
x Є N. Show that this function is
18. Let f : N U {O}  N U {O} be defined by
f (n) =
n+1
if n is even
n–1
if n is odd
Show that f is invertible and f = f –1
19. Let A = {1, 2, 3, 4, 5} and let f : A  A and g : A  A be defined as f(1) = 2,
f(2) = 3,
f(3) = 4, f(4) = 5, f(5) = 1 , g(1) = 4 , g(2) = 1, g(3) = 1, g(4) = 2, g(5) =
3. Find the
graphs of functions fog and gof.
20. Let f , g be real valued functions defined as f(x) = x2 + x + 7 ,
x Є R and
g(x) = 5x – 3 , x Є R. Find fog and gof. Also find (fog)(2) and (gof)(1).
21. Let f , g be real valued functions defined as
f (x) =
7x2 + x – 8
g(x) =
x≤1
4x + 5
1<x≤7
8x + 3
x>7
|x|
and
x < -3
0
-3≤ x <2
x2 + 4
x≥2
Find (fog)( -3) , (fog)(7) , (fog)(9) , (gof)(2) , (gof)(0) , (gof)(6)
22. Let f be a function defined on [0 , 1] defined by
f(x) =
if x Є Q
x
1–x
if x Є Q
Show that (fof)(x) = x
for x Є [0 , 1]
23. Let f(x) = ax / (x + 1) , x ≠ -1 . If (fof)(x) = x , find the value of a.
24. Let f : Z  Z and g : Z  Z be defined by f(n) = 3n and
g(n) =
n/3
if n is multiple of 3
0
if n is not a multiple of 3
for all n Є Z. Show that gof = IZ and fog ≠ IZ
25. Show that the function f : R  R defined by
xЄR
f (x) = (2x -1) / 3 ,
is one – one and onto function. Also find the inverse of the function f.
26. Let f : N  R be a function defined as
f (x) = 4x2 + 12x + 15.
Show that f : N  R is invertible
Binary Operations :
27. Let A = {1, 2, 3, 4, 5, 6} and * be an operation A defined by a * b = r , where
r is the least non – negative remainder when the product ab is divisible by 7.
Show that * is a binary operation on A.
28. Let * be a binary operation on the set IR defined by a * b = a + b + ab , a, b
Є R. Solve the equation
2 * (3 * x) = 7.
29. Let * be a binary operation defined by a * b = 2a + b – 3. Find 3 * 4
30. Consider the binary operation * on Q defined by a * b = a + 12b + ab for a, b
Є Q.
Find
(i) 2 * (1 / 3)
(ii) Show that * is not commutative
(iii) Show
that * is not associative.
31. Let * be a binary operation on N defined by a * b = 5ab
a, b Є IN. Discuss
the commutativity and associativity of the binary operation.
32. Let * be a binary operation on IN x IN defined by (a, b) * (c, d) = (ad + bc ,
bd) for
(a, b) , (c, d) C IN x IN. Discuss the commutativity and associativity
of this binary operation. Also find identity element, if any, in N x N.
33. Let A be a non empty set and let ‘*’ be a binary operation on P(A), the power
set A, defined by X * Y = (X – Y) U (Y – X), for X, Y Є P(A).
Show that (i) Ф Є P(A) is the identity element of (P(A) , * ).
(ii) X is invertible for
all X Є
P(A) and X –1 = X.
33. Let A = Q x Q. Let ‘*’ be a binary operation on A defined by (a, b) * (c, d) =
(ac, ad + b), then (i) find the identity element of (A, *) (ii) Find the invertible
elements of (A, *).
34. Let A be the set of all functions from a non-empty set S to itself. Discuss the
commutativity and associativity of the binary operation ‘Composition of
functions’ on the set A.
35. Let * be a binary operation on Z defined by a * b = a + b – 15 for a, b Є Z
i)
Show that * is commutative and associative
ii)
Find the identity element in (Z, *)
iii)
Find the inverse of an element in (Z, *)
36. Consider the binary operation * : Q x Q  Q defined by a * b = a + b – ab ;
a, b Є Q
i)
Is * commutative ?
ii)
Is * associative ?
iii)
Is identity for * exists ? If exists, find it.
iv)
Are elements of Q, invertible ? If so, find the inverse of a rational
number.
37. Let * be a binary operation on A = N x N defined by (a, b) * (c, d) = (a + c, b +
a, b, c, d Є N
d)
i)
Is * commutative ?
ii)
Is identity for * exist ? If exist find it.
iii)
Find inverse elements if exist.
38. The binary operations ‘*’ and ‘0’ on IR are defined by a * b = | a – b | and
a o b = a. Show that ‘*’ is commutative but not associative, ‘o’ is associative
but not commutative and ‘*’ is distributive over ‘o’.
39. Let A = N x N and * be a binary operation on A defined by (a, b) * (c, d) =
(ac, bd) for all a, b, c, d Є N. Show that ‘*’ is commutative and associative
binary operation on A.
40. Let ‘*’ be a binary operation on N defined by :
a * b = H.C.F of a and b.
Is ‘*’ commutative ? Is ‘*’ associative ?
Does there exist identity for this
binary operation on N?
Inverse Trignometric Functions
Find the principal value of each of the following :
i)
a. Cos-1 (1/2)
b. Tan-1 (1)
c. Sin-1 (1)
d. Cot-1 (0)
e. Cos-1 ( - 1/√2)
f. Sec-1 ( - √2)
g. Sin-1 ( - √3 / 2)
h. Cot-1 ( - √3)
i.
Cot-1 (√3)
j.
Cosec-1 ( -2 / √3)
k. Sin-1 ( 1 / √2)
l.
Tan-1 (0)
m. Cot-1 (1 / √3)
n. Sec-1 ( - 2 / √3)
o. Cosec-1 ( - √2)
INVERSE TRIGNOMETRY
Definitions and Formula :
1. Inverse of a function ‘f’ exist, if function is one – one and onto i.e. bijective.
2. Table for domain and range of Inverse Trignometric Function :
Function
Domain
y = sin-1 x
-1≤ x ≤ 1
Range (Principal value
Branch)
-Π/2 ≤ y ≤ Π/2
y = cos-1 x
-1≤x ≤1
0 ≤ y ≤ Π
y = tan-1 x
-∞<x <∞
-Π/2 < y < Π/2
x ≥ 1 or x ≤ -
y = cosec-1 x
x ≥ 1 or x ≤ -
y = sec-1 x
1
- Π / 2 ≤ y ≤ Π / 2, y
≠0
1
0≤y≤Π,
y≠ Π/2
y = cot-1 x
-∞ < x < ∞
0<y<Π
3. Properties of inverse trigonometric functions :
i)
Sin-1(sin x) = x,
x Є [ - Π / 2 , Π / 2]; sin(sin-1 x) = x ,
xЄ[-
1 , 1]
ii)
Cos-1(cos x) = x
cos(cos-1 x) = x
iii)
Tan-1(tan x) = x
tan(tan-1 x) = x
iv)
Cosec-1(cosec x) = x
cosec(cosec-1 x) = x
v)
Sec-1(sec x) = x
sec(sec-1 x) = x
vi)
Cot-1(cot x) = x
cot(cot-1 x) = x
4.
i)
x ≥ 1 or x ≤ -1 ;
Sin-1(1 / x) = cosec-1 x ,
cosec-1(1 / x)
= Sin-1 x
ii)
Tan-1(1 / x) = cot-1 x,
iii)
Sec-1(1 / x) = cos-1 x
cot-1(1 / x) = tan-1 x
x>0;
;cos-1(1 / x) = sec-1 x, x ≥ 1 or x ≤
-1
5.
i)
Sin-1x + Cos-1x = Π / 2
ii)
Tan-1x + Cot-1x = Π / 2
iii)
Sec-1x + Cosec-1x = Π / 2
4.
i)
Tan-1x + tan-1 y = tan-1 [( x + y) / (1 – xy)]
ii)
Tan-1x – tan-1 y = tan-1 [( x - y) / (1 + xy)] ,
6.
Sin-1 [2x / (1 + x2)] = cos-1 [(1 – x2) / (1 + x2)] = tan-1 [2x / (1 – x2)] = 2 tan-1x
7.
i)
Sin-1x + Sin-1y = Sin-1 (x √(1 – y2) + y √(1 – x2))
if xy < 1
xy > -1
ii)
Sin-1x – Sin-1y = Sin-1 (x √(1 – y2) - y √(1 – x2))
iii)
Cos-1x + Cos-1y = Cos-1 (xy - √(1 – x2) √(1 – y2))
iv)
Cos-1x – Cos-1y = Cos-1 (xy + √(1 – x2) √(1 – y2))
8.
While writing inverse trigonometrical functions in simplest form, we use the
following substitutions :
Term
Substitution
√(a2 – x2)
x = a sin θ or x = a cos θ
√(a2 + x2)
x = a tan θ or x = a cot θ
√(x2 – a2)
x = a sec θ or x = a cosec θ
√( a + x ) / √( a – x )
x = a cos θ
or x = a cos 2θ
INVERSE TRIGNOMETRIC FUNCTIONS :
1. Find the principal values of
(i)
sin-1 ½
(ii)
sec-1 (2 / √ 3)
(iii)
cosec-1 ( -1)
(iv)
cot-1 (-1 / √3)
2. Evaluate the following :
(i)
sin-1(sin (4Π / 3))
(ii)
cos-1(cos (2Π / 3))
(iii)
tan-1(tan (3Π / 4))
(iv)
cot-1(cot (-Π / 4))
(v)
sec-1(sec (5Π / 3))
(v)
cosec-1(cosec (3Π / 4))
(vi)
sin [(Π / 3) – sin-1( - ½ )]
(vii)
cos-1(cos (7Π / 6))
(viii)
cos [cos-1 ( -√ 3 / 2) + (Π / 6) ]
tan-1(tan – (3Π / 4))
(ix)
sin(cos-1 (3 / 5))
(x)
cos(tan-1 (3 / 4))
(xi)
sin (2 sin-1 0.8)
(xii)
tan ½ (cos-1 (√5 / 3))
3. Evaluate the following :
i)
Sin-1(sin 10)
ii)
Sin-1(sin 5)
iii)
Cos-1(cos 10)
iv)
Tan-1(tan (-6))
4. Prove that :
Cos-1x = 2 sin-1√((1 – x) / 2) = 2 cos-1√((1 + x) / 2)
5. Prove that : tan-1x = sin-1 (x / (√(1 + x2)) = cos-1 (1 / √(1 + x2)) = cot-1 (1 / x)
= cosec-1 √((1 + x2) / x) = sec-1 √(1 + x2)
6. Prove the following :
i)
Tan-1 (½) + tan-1 (⅓) = Π / 4
ii)
2 tan-1 (1 / 5) + tan-1 (¼) = tan-1 (32 / 43)
iii)
4 tan-1 (1 / 5) - tan-1 (1 / 70) + tan-1 (1 / 99) = Π / 4
iv)
2 tan-1 (1 / 5) + sec-1 (5√2 / 7) + 2 tan-1 (⅛) = Π / 4
v)
Tan-1 1 + tan-1 2 + tan-1 3 = 2 (tan-1 1 + tan-1 ½ + tan-1 ⅓)
vi)
Cos (cos-1 (15 / 17) - cos-1 (7 / 25)) = 297 / 425
vii)
Tan (2 sin-1 (4 / 5) + cos-1 (12 / 13)) = - 253 / 204
viii)
Sin-1 (3 / 5) + sin-1 (8 / 17) = sin-1 (77 / 85)
ix)
Sin-1 (1 / √5) + sin-1 (2 / √5) = Π / 2
x)
Sin-1 (3 / 5) + cos-1 (12 / 13) = cos-1 (33 / 65)
xi)
Sin-1 (4 / 5) + sin-1 (5 / 13) + sin-1 (16 / 65) = Π / 2
7. Show that
2 tan-1 (1 / 3) + tan-1 (1 / 7) = sec-1 (√34 / 5) + cosec-1 √17
8. Prove that :
2 tan-1 [√((a – b) / (a + b)) tan (θ / 2)] = cos-1 [(b + a cos θ) /
(a + b cos θ)]
9. If cos-1 x + cos-1 y + cos-1 z = Π and 0 < x , y, z < 1 , show that x2 + y2 +
z2 + 2xyz = 1
10. Prove that (i) tan-1 (x / √(a2 – x2) = sin-1 x / a
11. If - 1 ≤ x , y, z ≤ 1 , such that sin-1x + sin-1y + sin-1z = 3Π / 2 , find the value
of
x2000 + y2001 + z2002 – ( 9 / (x2000 + y2001 + z2002))
12. What is the principal value of : cos-1(cos 2Π / 3) + sin-1(sin 2Π / 3) ?
13. Prove that : tan-1(63 / 16) = sin-1(5 / 13) + cos-1(3 / 5)
14. Find the value of 2 tan-1(1 / 5) + sec-1(5√2 / 7) + 2 tan-1(1 / 8)
15. Write the range of one branch of sin-1 x, other than the principal branch.
16. Solve tan-1 2x + tan-1 3x = Π / 4
17. Sin (sin-1 (1 / 5) + cos-1 x ) = 1
18. Solve tan-1( x – 1 / x – 2) + tan-1 (x + 1 / x + 2) = Π / 4
19. Prove that tan-1 (1/3) + tan-1 (1/5) + tan-1 (1/7) + tan-1 (1/8) = Π / 4
( CBSE 2007)
20. Find the value of
(Pre- B-2007)
Cos-1 (1/2) + 3 sin-1(1/2)
21. If sin-1(2x / (1 + x2)) + sin-1(2y / (1 + y2)) = 2 tan-1 a , then find the value of a.
22. Find the value of tan-1 1 + Cot-1 ( -1) + Cos-1 (1/√2) + Sin-1(1/√2)
23. Prove that cot-1 7 + cot-1 8 + cot-1 18 = cot-1 3
24. What is the principal value of : cos-1(cos (Π / 3)) + sin-1(sin (Π / 3)) ?
25. Prove that sin-1 (8 / 17) + sin-1 (3 / 5) = tan-1 (77 / 66)
26. Prove that sin-1 (1 / 13) + cos-1 (4 / 5) + tan-1 (63 / 16) = Π
27. Evaluate sin [(Π / 3) – sin-1( - √3 / 4) ]
28. Prove that cot-1 (((√1 + sin x) + (√1 – sin x)) / ((√1 + sin x) – (√1 – sin x))) = x /
2,
x Є (0 , Π / 4)
29. If sin-1 (2a / (1+a2 )) – cos-1 ((1 – b2) / (1 + b2)) = tan-1 (2x / (1 – x2)) ,
prove that x = (a – b) / (1 + ab)
30. Solve : 3 sin-1 (2x / (1 + x2)) - 4 cos-1 [(1 – x2) / (1 + x2)] + 2 tan-1 (2x / (1 –
x2)) = Π / 3
31. Write tan-1 [ x / (a + (√a2 – x))] , - a < x < a in the simplest form.
32. If tan-1 { [√(1 + x2) – √(1 – x2)] / [√(1 + x2) + √(1 + x2)]} = α , then prove that
x2 = sin 2α
33. If sin-1 (2a / (1 + a2)) + sin-1 (2b / (1 + b2)) = 2 tan-1 x , prove that x = (a + b)
/ (1 – ab)
34. Write sin [ 2 tan-1 [√(1 – x) / (1 + x)]] in the simplest form.
35. Prove that sin {tan-1 (1)} = 1 / √2
36. Solve : tan-1((1 – x) / (1 + x)) – ½ tan-1 x = 0 ,
when x > 0
37. Find the principal value of tan-1( -1)
38. Prove that tan (cos-1 (4/5) + tan-1 (2/3)) = 17 / 6
39. Solve : tan-1 ((x – 1) / (x + 1)) + tan-1 ((2x – 1) / (2x + 1)) = tan-1 (23 / 36)
40. Show that : tan-1 √x = ½ cos-1 ((1 – x) / (1 + x))
41. ½ tan-1 x = cos-1 [ 1 + √(1 + x2 ) / 2√(1 + x2 )] ½
42. Show that tan-1 t + tan-1 (2t / (1 – t2 )) = tan-1 ((3t – t3) / (1 – 3t2)) , t2 < 1/3
43. Show that : tan-1 (m / n) - tan-1((m – n) / (m + n)) = Π / 4
44. Show that : tan-1 n + cot-1(n + 1) = tan-1 (n2 + n + 1)
45. Show that : tan-1 (2 / 11) + cot-1 (24 / 7) = tan-1 ½
46. If tan-1 x + tan-1 y + tan-1 z = Π , show that x + y + z = xyz
47. If cos-1 (x / a) + cos-1 (y / b) = α , show that (x2 / a2) – (2xy / ab) cos α + (y2
/ b2) = sin2 α
48. Show that cos ( 2 tan-1 (1/7)) = sin (4 tan-1 (1/3))
49. Solve the equation : tan-1 (1 / (1 + 2x)) + tan-1 (1 / (4x + 1)) = tan-1 (2 / x2)
50. Solve : tan-1 (2x / (1 – x2)) + cot-1 ((1 – x2) / 2x) = Π / 3 , x > 0
51. Solve : sin-1 (2a / (1 + a2 )) + sin-1 (2b / (1 + b2 )) = 2 tan-1 x , ab < 1
52. Prove that
tan-1 ((√x + √y) / (1 – √xy)) = tan-1 √x + tan-1 √y
53. Prove cot-1 ((ab + 1) / (a – b)) + cot-1 ((bc + 1) / (b – c)) + cot-1 ((ca + 1) / (c
– a)) = 0
54. Prove that tan-1 ((a – b) / (1 + ab)) + tan ((b – c) / (1 + bc)) + tan-1 ((c – a) /
(1 + ca)) = 0
55. If cos-1 (x / 2) + cos-1 (y / 3) = θ , prove that 9x2 – 12 xy cos θ + 4y2 = 36
sin2 θ