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.
317 exam-style questions on OCR A Level Maths 1.5 Trigonometry, covering 1.5.1 Definitions for all arguments, 1.5.2 Sine and cosine rules, 1.5.3 Area of a triangle, 1.5.4 Radian measure (A-level only), 1.5.5 Small angle approximations (A-level only), 1.5.6 Graphs of basic trigonometric functions, 1.5.7 Exact values in radians (A-level only), 1.5.8 Reciprocal and inverse trigonometric ratios (A-level only), 1.5.9 Graphs of reciprocal and inverse functions (A-level only), 1.5.10 Trigonometric identities, 1.5.11 Further trigonometric identities (A-level only), 1.5.12 Double angle and compound angle formulae (A-level only), 1.5.13 Geometrical proofs of formulae (A-level only), 1.5.14 Harmonic form Rcos / Rsin (A-level only), 1.5.15 Trigonometric equations, 1.5.16 Proof involving trigonometric functions (A-level only), and 1.5.17 Trigonometric functions in context. Each one has a worked solution and a mark scheme showing where the marks go.