Calculation based on sin inverse x and other trigonometric ratios

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Properties of inverse trigonometric function

\[sin^{-1}(-x)=-sin^{-1}(x)\]

\[ cos^{-1}(-x)=\pi -cos^{-1}(x)\]

\[ tan^{-1}(-x)=-tan^{-1}(x)\]

\[ cosec^{-1}(-x)=-cosec^{-1}(x)\]

\[ sec^{-1}(-x)=\pi -sec^{-1}(x)\]

\[ cot^{-1}(-x)=-cot^{-1}(x)\]

Properties of complementary angle

\[ sin(\frac{\pi }{2}-x)=cosx\]

\[cos(\frac{\pi }{2}-x)=sinx\]

\[tan(\frac{\pi }{2}-x)=cotx\]

\[cosec(\frac{\pi }{2}-x)=secx\]

\[sec(\frac{\pi }{2}-x)=cosecx\]

Calculation on invere function

\[sin(\frac{\pi }{2}-sin^{-1}(-\frac{\sqrt{3}}{2}))\]

\[sin(\frac{\pi }{2}+sin^{-1}(\frac{\sqrt{3}}{2}))\]

 \[sin(\frac{5\pi }{6})\]=\[ \frac{1}{2}\]