e) \( \cot \theta+\tan \theta=2 \csc 2 \theta \)

To verify the trigonometric identity: \[ \cot \theta + \tan \theta = 2 \csc 2\theta \] we'll start by expressing all terms in terms of sine and cosine functions and then simplify both sides. ### Step 1: Express \(\cot \theta\) and \(\tan \theta\) in Terms of Sine and Cosine Recall the definitions: \[ \cot \theta = \frac{\cos \theta}{\sin \theta} \quad \text{and} \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \] So, \[ \cot \theta + \tan \theta = \frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\cos \theta} \] ### Step 2: Combine the Two Fractions To add the fractions, find a common denominator, which is \(\sin \theta \cos \theta\): \[ \cot \theta + \tan \theta = \frac{\cos^2 \theta + \sin^2 \theta}{\sin \theta \cos \theta} \] ### Step 3: Simplify the Numerator Recall the Pythagorean identity: \[ \cos^2 \theta + \sin^2 \theta = 1 \] So, \[ \cot \theta + \tan \theta = \frac{1}{\sin \theta \cos \theta} \] ### Step 4: Express \(2 \csc 2\theta\) in Terms of Sine and Cosine First, recall that: \[ \csc 2\theta = \frac{1}{\sin 2\theta} \] and the double-angle identity for sine: \[ \sin 2\theta = 2 \sin \theta \cos \theta \] Therefore, \[ 2 \csc 2\theta = 2 \times \frac{1}{2 \sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta} \] ### Step 5: Compare Both Sides From Steps 3 and 4, we have: \[ \cot \theta + \tan \theta = \frac{1}{\sin \theta \cos \theta} = 2 \csc 2\theta \] ### Conclusion Both expressions simplify to the same form: \[ \cot \theta + \tan \theta = 2 \csc 2\theta \] Thus, the identity is verified.