Show then solve: secant/cosecant/cotangent identities

A question is this type if and only if it involves reciprocal trig functions (secθ, cosecθ, cotθ) and requires using identities such as 1+tan²θ=sec²θ or cosec²θ=1+cot²θ to rewrite as a quadratic, then solve.

2 questions · Standard +0.2

1.05o Trigonometric equations: solve in given intervals
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AQA C3 2006 January Q4
7 marks Moderate -0.3
4 It is given that \(2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x\).
  1. Show that the equation \(2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x\) can be written in the form $$2 \cot ^ { 2 } x + 5 \cot x - 3 = 0$$
  2. Hence show that \(\tan x = 2\) or \(\tan x = - \frac { 1 } { 3 }\).
  3. Hence, or otherwise, solve the equation \(2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x\), giving all values of \(x\) in radians to one decimal place in the interval \(- \pi < x \leqslant \pi\).
WJEC Unit 3 2018 June Q4
5 marks Standard +0.8
Solve the equation $$2\tan^2\theta + 2\tan\theta - \sec^2\theta = 2,$$ for values of \(\theta\) between \(0°\) and \(360°\). [5]