If sin⁻¹x = tan⁻¹y, show that 1/x² - 1/y² = 1. – Class 12 Mathematics

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Q. If sin⁻¹x = tan⁻¹y, show that 1/x² - 1/y² = 1.

West Bengal WbchseWest Bengal HS (WBCHSE) Board 2025Subjective· 2mImportance★★★★★

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Set both inverse-trig expressions equal to a common angle $\theta$ , then convert to a Pythagorean identity.

Let $\sin^{-1}x=\tan^{-1}y=\theta$ . Then $x=\sin\theta$ and $y=\tan\theta$ .

$\frac{1}{x^2}-\frac{1}{y^2} = \frac{1}{\sin^2\theta}-\frac{1}{\tan^2\theta} = \frac{1}{\sin^2\theta}-\frac{\cos^2\theta}{\sin^2\theta} = \frac{1-\cos^2\theta}{\sin^2\theta}$

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Q.If sin⁻¹x = tan⁻¹y, show that 1/x² - 1/y² = 1.

Concept understanding — Reciprocal and Negative-Argument Properties of Inverse Trigonometric Functions

Q.If sin⁻¹x = tan⁻¹y, show that 1/x² - 1/y² = 1.

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