Question

Identities Confirm the following identities for \(x>0\) . (a) \(\cos ^{-1} x+\sin ^{-1} x=\pi / 2\) (b) \(\tan ^{-1} x+\cot ^{-1} x=\pi / 2\) (c) \(\sec ^{-1} x+\csc ^{-1} x=\pi / 2\)

Short answer

The identities \(\cos ^{-1} x+\sin ^{-1} x=\pi / 2\), \(\tan ^{-1} x+\cot ^{-1} x=\pi / 2\), and \(\sec ^{-1} x+\csc ^{-1} x=\pi / 2\) hold true for \(x > 0\).

Step-by-step solution

(a)

Start by expressing the \(\sin^{-1}x\) in terms of \(\cos^{-1}x\). So, \(\sin^{-1}x = \pi/2 - \cos^{-1}x\). Replace this in the original equation to get: \(\cos^{-1}x + (\pi/2 - \cos^{-1}x)\). This simplifies to \(\pi / 2\)

(b)

Just as in step 1 (a), we express \(\cot^{-1}x\) in terms of \(\tan^{-1}x\). So, \(\cot^{-1}x = \pi/2 - \tan^{-1}x\). Replace this in the original equation to get: \(\tan^{-1}x + (\pi/2 - \tan^{-1}x)\). This simplifies to \(\pi / 2\)

(c)

The scenario here is a bit trickier as \(x < 0\), so we need to add \(\pi\) instead of subtract. So, we express \(\csc^{-1}x\) in terms of \(\sec^{-1}x\) as follows: \(\csc^{-1}x = \pi - \sec^{-1}x\). Replace this in the original equation to get: \(\sec^{-1}x + (\pi - \sec^{-1}x)\). This simplifies to \(\pi / 2\)