Given that tan θ = p, where p is a constant, p ≠ ±1 use standard trigonometric identities, to find in terms of p, (a) tan 2θ (b) cos θ (c) cot(θ − 45°) Write each answer in its simplest form. - Edexcel - A-Level Maths Pure - Question 3 - 2015 - Paper 3

To find (\cos \theta), we use the identity:

$\tan \theta = \frac{\sin \theta}{\cos \theta}$

This leads to:

$\tan^2 \theta + 1 = \sec^2 \theta$

Substituting (\tan \theta = p) gives:

$p^2 + 1 = \sec^2 \theta$

Thus, we have:

$\sec^2 \theta = 1 + p^2$

Therefore:

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

Using the identity for cotangent:

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

Knowing (\cot \theta = \frac{1}{\tan \theta} = \frac{1}{p}), we substitute:

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

Simplifying this:

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

Thus, the answer is:

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