Given that cosϕ≠±1\cos \phi \neq \pm 1cosϕ=±1, prove the identity
11−cosϕ+11+cosϕ≡2csc2ϕ \frac{1}{1 - \cos \phi} + \frac{1}{1 + \cos \phi} \equiv 2 \csc^2 \phi 1−cosϕ1+1+cosϕ1≡2csc2ϕIn a study of robotic arm resonance, the stability factor kkk is defined by the equation
11−cosϕ+11+cosϕ=k \frac{1}{1 - \cos \phi} + \frac{1}{1 + \cos \phi} = k 1−cosϕ1+1+cosϕ1=kDetermine the set of values of kkk for which this equation has real solutions for ϕ\phiϕ. Fully justify your answer.
Given that ϕ\phiϕ is in the third quadrant (reflex angle between 180∘180^\circ180∘ and 270∘270^\circ270∘) and
11−cosϕ+11+cosϕ=18 \frac{1}{1 - \cos \phi} + \frac{1}{1 + \cos \phi} = 18 1−cosϕ1+1+cosϕ1=18find the exact value of cotϕ\cot \phicotϕ.
Practise AQA A Level Maths 1.8 E: Trigonometry with exam-style questions for A Level Maths. 239 questions covering 1.8.1 Trigonometric definitions, rules and radians, 1.8.2 Small angle approximations (A-level only), 1.8.3 Trigonometric functions and exact values, 1.8.4 Reciprocal and inverse trigonometric functions (A-level only), 1.8.5 Trigonometric identities, 1.8.6 Compound and double angle formulae (A-level only), 1.8.7 Solving trigonometric equations, 1.8.8 Proofs with trigonometric identities, and 1.8.9 Trigonometry in context, matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
