Given that $$\tan\theta^0 = p$$, where $p$ is a constant, $p \neq \pm 1$ use standard trigonometric identities, to find in terms of $p$, (a) $\tan 2\theta^0$ (b) $\cos\theta^0$ (c) $\cot(\theta - 45^\circ)$ Write each answer in its simplest form. - Edexcel - A-Level Maths Pure - Question 2 - 2015 - Paper 3

To express $\cos\theta^0$ in terms of $p$, we can use the identity:

$\cos\theta = \frac{1}{\sqrt{1 + \tan^2\theta}}$

Substituting $\tan\theta = p$, we have:

$\cos\theta = \frac{1}{\sqrt{1 + p^2}}$

For $\cot(\theta - 45^\circ)$, we can use the identity:

$\cot(\theta - 45^\circ) = \frac{\cot\theta + 1}{1 - \cot\theta}$

Since $\cot\theta = \frac{1}{\tan\theta} = \frac{1}{p}$, we substitute:

$\cot(\theta - 45^\circ) = \frac{\frac{1}{p} + 1}{1 - \frac{1}{p}}$

This simplifies to:

$\frac{1 + p}{p - 1}$