Maths · Inverse trigonometrical functions and their properties
Assertion \) If then +\cos ^{-1}(\sin x)=\pi-2 x \) Reason \cos ^{-1} x=\frac{\p
Assertion \( (A) \) If \( 0<x<\frac{\pi}{2} \) then \( \sin ^{-1}(\cos x)+\cos ^{-1}(\sin x)=\pi-2 x \) Reason \( (\mathrm{R}) \cos ^{-1} x=\frac{\pi}{2}-\sin ^{-1} x \forall x \in \) \( [\mathbf{0}, \mathbf{1}] \)
- A. Both \( A \) and \( R \) are true and \( R \) is the correct explanation of \( A \)
- B. Both A and R are true but R is not correct explanation of \( A \)
- C. \( A \) is true but Ris false
- D. A is false but \( R \) is true
Step-by-step solution
For x in (0, π/2), using identities sin^{-1}(cos x) = π/2 - x and cos^{-1}(sin x) = π/2 - x, the sum becomes π - 2x. Reason R is the identity cos^{-1} x = π/2 - sin^{-1} x, which holds for x in [0,1] and can be used to derive both terms, thus correctly explains A.
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