Question

Prove that sin^-1(3/5)-sin^-1(8/17)=cos^-1(84/85).

Answer 1

Step-by-step explanation:

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Answer 2

Solutions

$\bf \large \: \sin(x) = \frac{3}{5} \: and \: \sin(y) = \frac{8}{17} \\ \cos(y) = \sqrt{1 - { \sin }^{2}x } \\ = \sqrt{1 - \frac{9}{25} } \\ = \frac{4}{5} \\ and \\ \cos(y) = \sqrt{1 - \sin^{2} y} \\ = \sqrt{1 - \frac{64}{289} } \\ = \frac{15}{17} \\ \\ now \: we \: have \: \\ \cos(x - y) = \cos(x) \cos(y) + \sin(x) \sin(y) \\ = \frac{4}{5} \times \frac{15}{17} + \frac{3}{5} \times \frac{8}{17} \\ = \frac{84}{85} \\ therefore \\ x - y = { \cos}^{ - 1} \frac{84}{85} \\ \\ hence \\ { \sin }^{ - 1} \frac{3}{5} - { \sin }^{ - 1} \frac{8}{17} = { \cos }^{ - 1} \frac{84}{85}$

Hence proved

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