Question

Evaluate \[ \frac{d}{dx}\left(\sin^{-1}(\tan x)+\tan^{-1}(\sin x)\right). \]

• A. \[ \frac{1}{\cos x\sqrt{\cos2x}} +\frac{2\cos x}{3-\cos2x} \]
• B. \[ \frac{\sec x}{\sqrt{1+\tan^2x}} +\frac{\cos x}{1-\sin^2x} \]
• C. \[ \frac{\sec x\tan x}{\sqrt{\cos2x}} +\frac{\sin x}{1+\cos2x} \]
• D. \[ \frac{\sec x}{1-\tan^2x} +\frac{\cos x}{\sqrt{1+\sin^2x}} \]

Hint:

While differentiating inverse trigonometric functions, first apply the chain rule and then simplify using identities such as \[ \boxed{1-\tan^2x=\frac{\cos2x}{\cos^2x}} \] and \[ \boxed{1+\sin^2x=\frac{3-\cos2x}{2}.} \]

Answer 1

The Correct Option is A