Question

Sin^-1 3/5+cosec^-1 5/4​

Answer 1

Question :

$\sf{\sin{}^{-1}\frac{3}{5}+\cosec{}^{-1}\frac{5}{4}}$

Formula's Used :

1) Conversion Property :

$\sf{\sin{}^{-1}x=\cosec{}^{-1}\dfrac{1}{x}}$

$\sf{\cosec{}^{-1}x=\sin{}^{-1}\dfrac{1}{x}}$

2) Property

$\sf{\sin{}^{-1}x+\sin{}^{-1}y=\sin{}^{-1}(x\sqrt{1-y^2}+y\sqrt{1-x^2})}$

Solution :

We have ,

$\sf{\sin{}^{-1}\frac{3}{5}+\cosec{}^{-1}\frac{5}{4}}$

$\sf{=\sin{}^{-1}\frac{3}{5}+\sin{}^{-1}\frac{4}{5}}$

$\sf{=\sin{}^{-1}[\frac{3}{5}\sqrt{1-(\frac{4}{5})^2}+\frac{4}{5}\sqrt{1-(\frac{3}{5})^2}]}$

$\sf{=\sin{}^{-1}[\frac{3}{5}\sqrt{1-\frac{16}{25}}+\frac{4}{5}\sqrt{1-\frac{9}{25}}]}$

$\sf{=\sin{}^{-1}[\frac{3}{5}\sqrt{\frac{25-16}{25}}+\frac{4}{5}\sqrt{\frac{25-9}{25}}]}$

$\sf{=\sin{}^{-1}[\frac{3}{5}\times\frac{3}{5}+\frac{4}{5}\times\frac{4}{5}]}$

$\sf{=\sin{}^{-1}[\frac{9}{25}+\frac{16}{25}]}$

$\sf{=\dfrac{\pi}{2}}$

Answer 2

Answer:

Refer to the attachment mate.......