TS Inter 1st Year Maths 1A Solutions Chapter 9 Hyperbolic Functions Ex 9(a)

Students must practice these TS Intermediate Maths 1A Solutions Chapter 9 Hyperbolic Functions Ex 9(a) to find a better approach to solving the problems.

TS Inter 1st Year Maths 1A Hyperbolic Functions Solutions Exercise 9(a)

I.
Question 1.
If sinh x = \(\frac{3}{4}\) find cosh (2x) and sinh (2x). (May 2014, Mar.’14, ’12)
Answer:
Given sinh x = \(\frac{3}{4}\)
and we have cosh2 x – sinh2 x = 1
TS Inter 1st Year Maths 1A Solutions Chapter 9 Hyperbola Ex 9(a) 1

TS Inter 1st Year Maths 1A Solutions Chapter 9 Hyperbolic Functions Ex 9(a)

Question 2.
If sinhx = 3, then show that
x = loge(3 + √10) (Board New Model Paper).
Answer:
Given sin hx = 3
⇒ x = sinh-1 3 = loge (3 + \(\sqrt{3^2+1}\))
= loge (3 + √10)
(∵ sinh x = log (x + \(\sqrt{x^2+1}\)) ∀ x ∈ R)

Question 3.
Prove that
(i) tanh (x – y) = \(\frac{\tanh x-\tanh y}{1-\tanh x \tanh y}\)
Answer:
TS Inter 1st Year Maths 1A Solutions Chapter 9 Hyperbola Ex 9(a) 2

(ii) coth (x – y) = \(\frac{{coth} x \cdot {coth} y-1}{{coth} y-{coth} x}\)
Answer:
TS Inter 1st Year Maths 1A Solutions Chapter 9 Hyperbola Ex 9(a) 3

Question 4.
Prove that
(i) (cosh x – sinh x)n
= cosh (nx) – sinh (nx) for any n ∈R.
Answer:
TS Inter 1st Year Maths 1A Solutions Chapter 9 Hyperbola Ex 9(a) 4

(ii) (cosh x + sinh x)n = cosh (nx) + sinh (nx) for any n ∈ R.
Answer:
TS Inter 1st Year Maths 1A Solutions Chapter 9 Hyperbola Ex 9(a) 5

TS Inter 1st Year Maths 1A Solutions Chapter 9 Hyperbolic Functions Ex 9(a)

Question 5.
Prove that
\(\frac{\tanh x}{{sech} x-1}+\frac{\tanh x}{{sech} x+1}\) = -2cosechx for x ≠ 0.
Answer:
TS Inter 1st Year Maths 1A Solutions Chapter 9 Hyperbola Ex 9(a) 6

Question 6.
Prove that \(\frac{\cosh x}{1-\tanh x}+\frac{\sinh x}{1-{coth} x}\) = sinhx + coshx for x ≠ 0.
Answer:
TS Inter 1st Year Maths 1A Solutions Chapter 9 Hyperbola Ex 9(a) 7

Question 7.
For any x ∈ R, prove that cosh4 x – sinh4 x = cosh 2x.
Answer:
cosh4 x – sinh4 x
= (cosh2 x)2 – (sinh2 x)2
= (cosh2 x + sinh2 x) (cosh2 x – sinh2 x)
= cosh 2x (1) = cosh 2x

TS Inter 1st Year Maths 1A Solutions Chapter 9 Hyperbolic Functions Ex 9(a)

Question 8.
If u = loge \(\left\{\tan \left(\frac{\pi}{4}+\frac{\theta}{2}\right)\right\}\) and if cos θ > 0 then prove that cosh u = sec θ
Answer:
TS Inter 1st Year Maths 1A Solutions Chapter 9 Hyperbola Ex 9(a) 8

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