3.5 Property 5

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) ${\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \frac{\pi }{2}; - 1 \leqslant x \leqslant 1$

Proof: Put $$\begin{equation} \begin{aligned} {\sin ^{ - 1}}x = A \\ \Rightarrow x = \sin A \\ {\cos ^{ - 1}}x = B \\ x = \cos B \\ \sin A = \cos B \\ \sin A = \sin \left( {\frac{\pi }{2} - B} \right) \\\end{aligned} \end{equation} $$where $\left( {A,\left( {\frac{\pi }{2} - B} \right) \in \left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right]} \right)$

$$\begin{equation} \begin{aligned} A = \frac{\pi }{2} - B \\ A + B = \frac{\pi }{2} \\\end{aligned} \end{equation} $$

Putting the values of $A$ and $B$ we get $${\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \frac{\pi }{2}$$

(ii) ${\tan ^{ - 1}}x + {\cot ^{ - 1}}x = \frac{\pi }{2};x \in R$

(iii) ${\text{cose}}{{\text{c}}^{ - 1}}x + {\sec ^{ - 1}}x = \frac{\pi }{2};\;\left| x \right| \geqslant 1$