Practicing the Intermediate 1st Year Maths 1A Textbook Solutions Inter 1st Year Maths 1A Hyperbolic Functions Solutions Exercise 9(a) will help students to clear their doubts quickly.

## Intermediate 1st Year Maths 1A Hyperbolic Functions Solutions Exercise 9(a)

Question 1.
If sinh x = $$\frac{3}{4}$$, find cosh (2x) and sinh (2x).
Solution:

Question 2.
If sinh x = 3, then show that x = loge(3 + √10).
Solution:

Question 3.
Prove that
(i) tanh (x – y) = $$\frac{\tanh x-\tanh y}{1-\tanh x \tanh y}$$
Solution:

(ii) coth (x – y) = $$\frac{{coth} x \cdot {coth} y-1}{{coth} y-{coth} x}$$
Solution:

Question 4.
Prove that
(i) (cosh x – sinh x)n = cosh (nx) – sinh (nx), for any n ∈ R.
Solution:

(ii) (cosh x + sinh x)n = cosh (nx) + sinh (nx), for any n ∈ R.
Solution:

Question 5.
Prove that $$\frac{\tanh x}{{sech} x-1}+\frac{\tanh x}{{sech} x+1}$$ = -2 cosech x, for x ≠ 0
Solution:

Question 6.
Prove that $$\frac{\cosh x}{1-\tanh x}+\frac{\sinh x}{1-{coth} x}$$ = sinh x + cosh x, for x ≠ 0
Solution:

Question 7.
For any x ∈ R, prove that cosh4x – sinh4x = cosh (2x)
Solution:
L.H.S = cosh4x – sinh4x
= (cosh2x)2 – (sinh2x)2
= [cosh2x – sinh2x] [cosh2x + sinh2x]
= (1) cosh (2x)
= cosh (2x)
∴ cosh4x – sinh4x = cosh (2x)

Question 8.
If u = $$\log _{e}\left(\tan \left(\frac{\pi}{4}+\frac{\theta}{2}\right)\right)$$ and if cos θ > 0,then prove that cosh u = sec θ.
Solution: