3.3 Property 3

2.1 ${\sin ^{ - 1}}x$:
2.2 ${\cos ^{ - 1}}x$:
2.3 ${\tan ^{ - 1}}x$:
2.4 ${\text{cose}}{{\text{c}}^{ - 1}}x$:
2.5 ${\sec^{ - 1}}x$
2.6 ${\cot^{ - 1}}x$
2.7 Summary

3.1 Property 1
3.2 Property 2
3.3 Property 3
3.4 Property 4
3.5 Property 5
3.6 Property 6
3.7 Property 7
3.8 Property 8
3.9 Property 9
3.10 Property 10
3.11 Property 11
3.12 Property 12
3.13 Property 13

(i) $\sin \left( {{{\sin }^{ - 1}}x} \right) = x$ where $ - 1 \leqslant x \leqslant 1$

Proof: Let $$\theta = {\sin ^{ - 1}}x...(i)$$

$$\begin{equation} \begin{aligned} \theta \in \left[ {\frac{{ - \pi }}{2},\frac{\pi }{2}} \right] \\ \frac{{ - \pi }}{2} \leqslant {\sin ^{ - 1}}x \leqslant \frac{\pi }{2} \\ - 1 \leqslant x \leqslant 1 \\ \sin \theta = x \\\end{aligned} \end{equation} $$

Put the value of $\theta $ from equation $(i)$, we get

$$\sin \left( {{{\sin }^{ - 1}}x} \right) = x$$

(ii) $\cos \left( {{{\cos }^{ - 1}}x} \right) = x; - 1 \leqslant x \leqslant 1$

(iii) $\tan \left( {{{\tan }^{ - 1}}x} \right) = x;x \in R$

(iv) ${\text{cosec}}\left( {{\text{cose}}{{\text{c}}^{ - 1}}x} \right) = x;\left| x \right| > 1$

(v) $\sec \left( {{{\sec }^{ - 1}}x} \right) = x;x \leqslant - 1,x \geqslant 1$

(vi) $\cot \left( {{{\cot }^{ - 1}}x} \right) = x;x \in R$