(a) Prove that $$ tan\theta + cot\theta = 2cosec2\theta, \quad \theta \neq \frac{n\pi}{2}, \quad n \in \mathbb{Z} - Edexcel - A-Level Maths Pure - Question 10 - 2017 - Paper 1

To prove the identity, we start by rewriting the left-hand side:

1. Write the definitions of tangent and cotangent: $tan\theta = \frac{sin\theta}{cos\theta}, \quad cot\theta = \frac{cos\theta}{sin\theta}$
2. Combine the two terms: $tan\theta + cot\theta = \frac{sin\theta}{cos\theta} + \frac{cos\theta}{sin\theta}$ $= \frac{sin^2\theta + cos^2\theta}{sin\theta cos\theta}$
3. Recognize that $sin^2\theta + cos^2\theta = 1$: $= \frac{1}{sin\theta cos\theta}$
4. Using the double angle identity, we know: $sin 2\theta = 2sin\theta cos\theta$ So, $\frac{1}{sin\theta cos\theta} = \frac{2}{sin 2\theta}$
5. Substituting this into the equation: $tan\theta + cot\theta = 2cosec2\theta$ This proves part (a) of the question.