Class12 ISC Chapter ITF Exercise 1.2

Class XII ISC Chapter ITF Exercise 1.2

Inverse trigonometric functions are a key topic in class 12 ISC Maths, providing the foundation for solving complex equations involving angles and trigonometric ratios. On this page, you'll find a comprehensive collection of practice questions covering various aspects of inverse trigonometric functions, including basic concepts, properties, and applications. These questions are designed to help students reinforce their understanding and build confidence in tackling problems related to sin^-1(𝑥),cos^-1(𝑥),tan^-1(𝑥) and more.
Each question is carefully crafted to align with the ISC syllabus, ensuring that you're well-prepared for your exams.

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ITF

Class XII ISC Chapter ITF Exercise 1.2

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Q1 Evaluate the following :

(i) sin-1(sin2)

(ii) sin-1(sin5)

(iii) tan-1(tan4)

(iv) tan-1(tan(-4))

(v) cos-1(cos 10)

Q2 Prove the following :

(i) tan-1( 1+x / 1-x )= π / 4 +tan-1 x,x< 1

(ii) tan-1 x+cot-1 (x+1) =tan-1(x2+x+1)

Q3 Prove the following (for suitable values of x and y) :

(i) tan-1 √x + tan-1 √y =tan-1 √ x + √ y / 1- √ xy

(ii) tan-1 x+ √x / 1-x3/2 = tan-1 x + tan-1 √x

(iii) tan-1( 1 / √3 tan x / 2 )= 1 / 2 cos-1 ( 1+2cosx / 2+cosx )

Q4 Write the following functions in simplest form :

(i) tan-1 ( √( 1-cosx / 1+cosx ))

(ii) tan-1 ( cosx-sinx / cosx+sinx )

(iii) tan-1 ( 2 √x / 1-x )

(iv) sin-1 ( √ ( x / 1+x ))

(v) tan-1 ( √ ( 1-x / 1+x ))

(vi) cos-1 (1-2x2)

(vii) sec-1 ( 1 / 2x2-1 )

(viii) tan-1 x / √ a2-x2

(Q5) Prove that : sin(cot-1(cos(tan-1x))) = √ x2 +1 / x2 +2

(Q6) Find the value of tan 1 / 2 (sin-1 2x / 1+x2 +cos-1 1-y2 / 1+y2 ),|x|< 1,y> 0,xy< 1

(Q7) Prove the following :

(i) tan-1 1 / 2 + tan-1 2 / 11 = tan-1 3 / 4

(ii) tan-1 1 / 7 + tan-1 1 / 13 = tan-1 2 / 9

(iii) tan-12- tan-11= tan-1 1 / 3

(iv) tan-11+ tan-1 1 / 2 +tan-1 1 / 3 = π / 2

(v) 2 tan-1 1 / 3 +tan-1 1 / 7 = π / 4

(vi) 2 tan-1 1 / 3 +cot-14= tan-1 16 / 13

(vii) tan-1 3 / 4 + tan-1 3 / 5 - tan-1 8 / 19 = π / 4

(viii) cot-11 + cot-12 +cot-13= π / 2

(ix) tan-12 + tan-13 = 3π / 4

(x) tan-1 1 / 5 + tan-1 1 / 7 + tan-1 1 / 3 +tan-1 1 / 8 = π / 4

(Q8) Prove the following :

(i) sin-1 1 / √5 + sin-1 2 / √5 = π / 2

(ii) cos-1 4 / 5 + cos-1 12 / 13 = cos-1 33 / 65

(iii) cos-1 3 / 5 + cos-1 12 / 13 = cos-1 56 / 65

(iv) tan-1 1 / 3 + sec-1 √ 5 / 2 = π / 4

(v) cos-1 4 / 5 + tan-1 3 / 5 = tan-1 27 / 11

(vi) cos-1 5 / √ 41 + cot-1 4 / 5 = π / 2

(vii) sin-1 4 / 5 + cos-1 2 / √5 = cot-1 2 / 11

(Q9) Evaluate : tan(2 tan-1 1 / 2 - cot-13)

(Q10) Find the values of the following :

(i) tan (cos-1 3 / 5 +tan-1 1 / 4 )

(ii) cos(sin-1 3 / 5 +sin-1 5 / 13 )

(iii) cot(cos-1 4 / 5 +sin-1 2 / √13 )

(iv) sin(2 tan-1 2 / 3 )+ cos( tan-1 √3)

(Q11) Solve the following equations for x :

(i) cos-1 (sin(cos-1 x)) = π / 6

(Q12) Solve the following equations for x :

(i) tan-1 (x+1) +tan-1 (x-1) = tan-1 6 / 17

(ii) tan-1 ( x-2 / x-3 ) + tan-1 ( x+2 / x+3 ) = π / 4

(iii) tan-1 ( x-2 / x-4 ) + tan-1 ( x+2 / x+4 ) = π / 4

(iv) cos (sin-1 x) = 1 / 9

(v) cos (2 sin-1 x)= 1 / 9

(vi) sin-1 8 / x + sin-1 15 / x = π / 2

(vii) sin-1 x+ sin-1 (1-x) = cos-1 x,x ≠0

(viii) sin-1 2a / 1+a2 + sin-1 2b / 1+b2 = 2 tan-1 x

(Q13) Find the values of:

(i) sin (sin-1x+ cos-1x), |x|≤ 1

(ii) cot (tan-1x+ cot-1x), x ∈ R

(Q14) Find the values of:

(i) cosec (sin-1x+ cos-1x), |x|≤ 1

(ii) cos (sec-1x+ cosec-1x), |x|≥ 1

(Q15) Find the values of:

(i) tan-1 (sin(- π / 2 ))

(ii) cos (tan-1 3 / 4 )

(iii) tan (cos-1 8 / 17 )

(Q16) Simplify the following :

(i) tan-1( sinx / 1+cosx )

(ii) tan-1( x / √ 1-x2 ),|x| < 1

(Q17) Solve the following equations for x :

(i) tan-1x = sin-1 ( 1 / √2 )

(ii) cos(sin-1x)= 1 / 2

(iii) 4 sin-1x + cos-1x = π

(iv) sin-1x - cos-1x = π / 6

(i) sin^-1(sin2)

(ii) sin^-1(sin5)

(iii) tan^-1(tan4)

(iv) tan^-1(tan(-4))

(v) cos^-1(cos 10)

(i) tan^-1(
1+x / 1-x
)=
π / 4
+tan^-1 x,x< 1

(ii) tan^-1 x+cot^-1 (x+1) =tan^-1(x²+x+1)

(i) tan^-1 √x + tan^-1 √y =tan^-1
√ x + √ y / 1- √ xy

(ii) tan^-1
x+ √x / 1-x^3/2
= tan^-1 x + tan^-1 √x

(iii) tan^-1(
1 / √3
tan
x / 2
)=
1 / 2
cos^-1 (
1+2cosx / 2+cosx
)

(i) tan^-1 ( √(
1-cosx / 1+cosx
))

(ii) tan^-1 (
cosx-sinx / cosx+sinx
)

(iii) tan^-1 (
2 √x / 1-x
)

(iv) sin^-1 ( √ (
x / 1+x
))

(v) tan^-1 ( √ (
1-x / 1+x
))

(vi) cos^-1 (1-2x²)

(vii) sec^-1 (
1 / 2x²-1
)

(viii) tan^-1
x / √ a²-x²

(Q5) Prove that : sin(cot^-1(cos(tan^-1x))) = √
x² +1 / x² +2

(Q6) Find the value of tan
1 / 2
(sin^-1
2x / 1+x²
+cos^-1
1-y² / 1+y²
),|x|< 1,y> 0,xy< 1

(i) tan^-1
1 / 2
+ tan^-1
2 / 11
= tan^-1
3 / 4

(ii) tan^-1
1 / 7
+ tan^-1
1 / 13
= tan^-1
2 / 9

(iii) tan^-12- tan^-11= tan^-1
1 / 3

(iv) tan^-11+ tan^-1
1 / 2
+tan^-1
1 / 3
=
π / 2

(v) 2 tan^-1
1 / 3
+tan^-1
1 / 7
=
π / 4

(vi) 2 tan^-1
1 / 3
+cot^-14= tan^-1
16 / 13

(vii) tan^-1
3 / 4
+ tan^-1
3 / 5
- tan^-1
8 / 19
=
π / 4

(viii) cot^-11 + cot^-12 +cot^-13=
π / 2

(ix) tan^-12 + tan^-13 =
3π / 4

(x) tan^-1
1 / 5
+ tan^-1
1 / 7
+ tan^-1
1 / 3
+tan^-1
1 / 8
=
π / 4

(i) sin^-1
1 / √5
+ sin^-1
2 / √5
=
π / 2

(ii) cos^-1
4 / 5
+ cos^-1
12 / 13
= cos^-1
33 / 65

(iii) cos^-1
3 / 5
+ cos^-1
12 / 13
= cos^-1
56 / 65

(iv) tan^-1
1 / 3
+ sec^-1
√ 5 / 2
=
π / 4

(v) cos^-1
4 / 5
+ tan^-1
3 / 5
= tan^-1
27 / 11

(vi) cos^-1
5 / √ 41
+ cot^-1
4 / 5
=
π / 2

(vii) sin^-1
4 / 5
+ cos^-1
2 / √5
= cot^-1
2 / 11

(Q9) Evaluate : tan(2 tan^-1
1 / 2
- cot^-13)

(i) tan (cos^-1
3 / 5
+tan^-1
1 / 4
)

(ii) cos(sin^-1
3 / 5
+sin^-1
5 / 13
)

(iii) cot(cos^-1
4 / 5
+sin^-1
2 / √13
)

(iv) sin(2 tan^-1
2 / 3
)+ cos( tan^-1 √3)

(i) cos^-1 (sin(cos^-1 x)) =
π / 6

(i) tan^-1 (x+1) +tan^-1 (x-1) = tan^-1
6 / 17

(ii) tan^-1 (
x-2 / x-3
) + tan^-1 (
x+2 / x+3
) =
π / 4

(iii) tan^-1 (
x-2 / x-4
) + tan^-1 (
x+2 / x+4
) =
π / 4

(iv) cos (sin^-1 x) =
1 / 9

(v) cos (2 sin^-1 x)=
1 / 9

(vi) sin^-1
8 / x
+ sin^-1
15 / x
=
π / 2

(vii) sin^-1 x+ sin^-1 (1-x) = cos^-1 x,x ≠0

(viii) sin^-1
2a / 1+a²
+ sin^-1
2b / 1+b²
= 2 tan^-1 x

(i) sin (sin^-1x+ cos^-1x), |x|≤ 1

(ii) cot (tan^-1x+ cot^-1x), x ∈ R

(i) cosec (sin^-1x+ cos^-1x), |x|≤ 1

(ii) cos (sec^-1x+ cosec^-1x), |x|≥ 1

(i) tan^-1 (sin(-
π / 2
))

(ii) cos (tan^-1
3 / 4
)

(iii) tan (cos^-1
8 / 17
)

(i) tan^-1(
sinx / 1+cosx
)

(ii) tan^-1(
x / √ 1-x²
),|x| < 1

(i) tan^-1x = sin^-1 (
1 / √2
)

(ii) cos(sin^-1x)=
1 / 2

(iii) 4 sin^-1x + cos^-1x = π

(iv) sin^-1x - cos^-1x =
π / 6

(v) 3 tan^-1 x+ cot^-1 x=π

(i) tan^-1
1 / 2
+ tan^-1
1 / 3

(ii) cot^-1 2 + cot^-1 3

Q19(i) If tan^-1 x =
π / 10
for some x∈R, find the value of cot^-1x

(ii) If sin^-1 x +sin^-1 y=
π / 2
,then find the value of cos^-1 x + cos^-1 y