Question

If $\sin^{-1} x = \tan^{-1 }y,$ what is the value of $ \frac{1}{x^{2}} - \frac{1}{y^{2}} $ ?

• A. 1
• B. -1
• C. 0
• D. 2

Answer 1

Answer: (A)

Let, $\sin^{-1} x = \tan^{-1 }y = \theta $
$ \Rightarrow x = \sin\theta\, y = \tan\theta$
$ \frac{1}{x^{2}} = \frac{1}{\sin^{2} \theta} = cosec ^{2} \theta$
and $ \frac{1}{y^{2}} = \frac{1}{\tan^{2} \theta} = \cot^{2} \theta . $
$ \Rightarrow \frac{1}{x^{2} } - \frac{1}{y^{2}} = cosec ^{2} \theta - \cot^{2} \theta = 1 $