Consider the following:<br>1. \sqrt{\sec^2\theta+\csc^2\theta}=\tan \theta+\cot \theta, where 0 \lt \theta \lt 90^\circ<br>2. \sqrt{\tan^2\theta+\cot^2\theta+4}=\sec \theta+\csc \theta, where 0 \lt \theta \lt 90^\circ<br>Which of the above is/are an identity/identities?

Explanation

For Statement 1, \sec^2\theta+\csc^2\theta = \frac{1}{\cos^2\theta}+\frac{1}{\sin^2\theta} = \frac{1}{\sin^2\theta\cos^2\theta}. Its square root is \frac{1}{\sin\theta\cos\theta} = \frac{\sin^2\theta+\cos^2\theta}{\sin\theta\cos\theta} = \tan\theta+\cot\theta, so it is correct. For Statement 2, trying \theta=45^\circ gives LHS =\sqrt{1+1+4}=\sqrt{6} and RHS =\sqrt{2}+\sqrt{2}=\sqrt{8}. They do not match. Only Statement 1 is an identity.

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