MHT CET · Maths · Inverse Trigonometric Functions
If \(\sin ^{-1}\left(\frac{x}{13}\right)+\operatorname{cosec}^{-1}\left(\frac{13}{12}\right)=\frac{\pi}{2}\), then the value of \(x\) is
- A 4
- B 12
- C 5
- D 11
Answer & Solution
Correct Answer
(C) 5
Step-by-step Solution
Detailed explanation
\(\begin{aligned}
& \sin ^{-1}\left(\frac{x}{13}\right)=\frac{\pi}{2}-\operatorname{cosec}^{-1}\left(\frac{13}{12}\right) \\
& =\sec ^{-1}\left(\frac{13}{12}\right)=\cos ^{-1}\left(\frac{12}{13}\right) \\
& \therefore \quad \sin ^{-1}\left(\frac{x}{13}\right)=\sin ^{-1}\left(\frac{5}{13}\right) \\
& \therefore \quad x=5
\end{aligned}\)
& \sin ^{-1}\left(\frac{x}{13}\right)=\frac{\pi}{2}-\operatorname{cosec}^{-1}\left(\frac{13}{12}\right) \\
& =\sec ^{-1}\left(\frac{13}{12}\right)=\cos ^{-1}\left(\frac{12}{13}\right) \\
& \therefore \quad \sin ^{-1}\left(\frac{x}{13}\right)=\sin ^{-1}\left(\frac{5}{13}\right) \\
& \therefore \quad x=5
\end{aligned}\)
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