Question

prove that 2tan-¹(1/2) + sin-¹(3/5) =π/2​

Answer 1

Step-by-step explanation:

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Answer 2

$2 \tan {}^{ - 1} ( \frac{1}{2} ) + \sin {}^{ - 1} ( \frac{3}{5} )$

$= 2 \tan {}^{ - 1} ( \frac{1}{2} ) + \tan {}^{ - 1} ( \frac{3}{ \sqrt{ {5}^{2} - {3}^{2} } } )$

As,

$\tan {}^{ - 1} (x) + \tan {}^{ -1 } (y) = \tan {}^{ - 1} ( \frac{x + y}{1 - xy} )$

$\implies \tan {}^{ - 1} ( \frac{ \frac{1}{2} + \frac{1}{2} }{1 - \frac{1}{2} \times \frac{1}{2} } ) + \tan {}^{ - 1} ( \frac{3}{4} )$

$= \tan {}^{ - 1} ( \frac{4}{3} ) + \tan {}^{ - 1} ( \frac{3}{4} )$

$= \tan {}^{ - 1} ( \frac{ \frac{4}{3} + \frac{3}{4} }{1 - \frac{3}{4} \times \frac{4}{3} } )$

$=\tan {}^{ - 1} ( \frac{ \frac{4}{3} + \frac{3}{4} }{0} ) = \frac{\pi}{2}$