If [a= ^-1( (5))] and [b= ^-1( (5))], then a^2+b^2 is equal to

Question

If $$a=\sin ^{-1}(\sin (5))$$ and $$b=\cos ^{-1}(\cos (5))$$, then $a^2+b^2$ is equal to

Accepted Answer

$$\begin{aligned}
& a=\sin ^{-1}(\sin 5)=5-2 \pi \
& \text { and } b=\cos ^{-1}(\cos 5)=2 \pi-5 \
& \therefore a^2+b^2=(5-2 \pi)^2+(2 \pi-5)^2 \
& =8 \pi^2-40 \pi+50
\end{aligned}$$

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