CHAPTER 2
DAV CENTENARY PUBLIC SCHOOL, PASCHIM ENCLAVE, NEW DELHI-87
CLASS-XII
SESSION 2016-17
Inverse trigonometric function.Expected marks : 5 (1 + 4)
Expected No. of Questions : 2
One Mark Questions
Q1. Write the principle value of following.
7𝜋
3𝜋
−1
a) cos-1cos 6
b) sin-1 sin 5
c) cos -1 ( 2 )
3𝜋
−𝜋
e)sin-1tan 4
f) cos-1cos 4
d) tan-1(-1)
−𝜋
g) sec2(tan-12)
h) cot-1 cot
4
1
Q2. If sin { sin-15 + cos-1 x } = 1 . Find x
Q3. Evaluate : a) sin[
𝜋
3
−1
2𝜋
- sin-1( 2 ) ]
2𝜋
b) cos-1cos ( 3 ) + sin-1sin( 3 )
Q4. Write one branch of tan-1x other than principle branch.
Q5. Simplify the following expressions:
a) tan-1
1−𝑐𝑜𝑠𝑥
𝑎𝑏 +1
1+𝑐𝑜𝑠𝑥
𝑥
𝑥−𝑦
𝑥−1
c) tan-1𝑦 - tan-1𝑥+𝑦
e) tan-1
𝑏𝑐 +1
𝑎𝑐 +1
b) cot-1 𝑎 −𝑏 + cot-1 𝑏−𝑐 + cot-1 𝑎−𝑐
𝑥+1
d) tan-1𝑥+1
𝑥
+ tan-1𝑥−1
f) cos 2tan-1
𝑎 2 −𝑥 2
1−𝑥
1+𝑥
Four marks questions
Q1. Prove the following:
−4
12
a) cos tan-1 3 + sin-1 13
=
63
1
1
1
b) 4 tan-1 5 - tan-1 70 + tan-199 =
65
𝜋
c) tan-1 1 + tan-1 2 + tan-1 3 = 2( cot-1 1 + cot-1 2 + cot-1 3)
4
5
16
d) sin-1 5 + sin-1 13 + sin-165 =
4
1
𝜋
f) sin-1 5 + 2tan-1 3 = 2
𝜋
2
1
2
12
4
1
3
e) tan-1 4 + tan-1 9 = 2cos-1 5
63
g) sin-1 13 + cos-1 5 +tan-116 = 𝜋
4
Q2. Draw the graph of a) f(x)= tan-1 x
b) g(x) = cosec-1x.
Q3. Simplify the following inverse trigonometric expressions:
𝑐𝑜𝑠𝑥
a) cot -1( 1 + 𝑥 2 + 𝑥)
𝜋
1
𝑎
c) tan ( 4 + 2cos-1 𝑏 )
3𝑥−4 1−𝑥 2
d) sin-1
5
f) cosec-1
𝑎 2 +𝑥 2
𝑥
b) tan-11+𝑠𝑖𝑛𝑥
𝜋
1
𝑎
+ tan ( 4 - 2cos-1 𝑏 )
𝑥
e) sin-1
9+𝑥 2
g) tan-1
.
1+𝑥 2 + 1−𝑥 2
1+𝑥 2 − 1−𝑥 2
Q4. Solve for x :
1
𝑥−1
a) tan(cos-1x) = sin(cot -1 2)
𝑥 2 −1
2𝑥
c) cos-1𝑥 2 +1 + tan-1𝑥 2 −1 =
2𝜋
1+𝑥
d) tan-11−𝑥 =
3
𝜋
1+𝑥 2 + 1
tan-1x = cos-1
2
2 1+𝑥 2
4𝑥(1−𝑥 2 )
Q6. Show that 4tan-1x = tan-11−6𝑥 2 +𝑥 4
𝑥
𝑦
𝑥−𝑦
𝑥+𝑦
𝜋
4
+ tan-1x
f) tan-1(x+1) + tan-1(x-1) = tan-1 31
1
Q7. Show that 2tan-1
𝜋
8
e) sin-1(1-x) – 2 sin-1x = 2
Q5. Show that
𝑥+1
b) tan -1𝑥−2 + tan-1𝑥+2 = 4
𝜃
𝑦 +𝑥𝑐𝑜𝑠𝜃
tan 2 = cos-1𝑥+𝑦𝑐𝑜𝑠𝜃
Q8. If cos-1𝑎 + cos-1 𝑏 = z. Show that
𝑥2
𝑎2
-
2𝑥𝑦
𝑎𝑏
𝑦2
cosz + 𝑏 2 = sin2z .
𝑥
Q9. If y = cot-1 𝑐𝑜𝑠𝑥 - tan-1 𝑐𝑜𝑠𝑥. Prove that sin y = tan2 2