If a = sin^−1(sin(5)) and b = cos−1(cos(5)), then a^2 + b^2 is equal to

Question

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

(1) \(4 \pi^{2}+25\)

(2) \(8 \pi^{2}-40 \pi+50\)

(3) \(4 \pi^{2}-20 \pi+50\)

(4) \(25\)

Answer 1

Correct option is (2) \(8 \pi^{2}-40 \pi+50\)

\( \mathrm{a}=\sin ^{-1}(\sin 5)=5-2 \pi\)

and \(b=\cos ^{-1}(\cos 5)=2 \pi-5\)

\(\therefore \mathrm{a}^{2}+\mathrm{b}^{2}=(5-2 \pi)^{2}+(2 \pi-5)^{2}\)

\(=8 \pi^{2}-40 \pi+50\)