Question

Prove that $sin[cot^{-1}\frac{2x}{1 - x^{2}} + cos^{-1}\frac{1 - x^{2}}{1 + x^{2}}] = 1$.

Answer 1

Answer:

1

Step-by-step explanation:

Formula used:

1. cos2A=(1-tan²A)/(1+tan²A)

2.tan2A = 2 tanA/(1+tan²A)

cot⁻¹(2x/(1-x²))

take x = tanA

cot⁻¹(2x/(1-x²))

= cot⁻¹(2tanA/(1-tan²A))

= cot⁻¹(tan2A)

= cot⁻¹(cot(90°-2A)

= 90°- 2A

cos⁻¹((1-x²)/(1+x²))

=cos⁻¹((1-tan²A)/(1+tan²A))

=cos⁻¹(cos2A)

=2A

Now

sin[cot⁻¹(2x/(1-x²)) + cos⁻¹((1-x²)/(1+x²))]

= sin [ 90°- 2A +2A]

= sin90°

=1