Prove that
tanϕ+cotϕ≡2csc2ϕ \tan \phi + \cot \phi \equiv 2 \csc 2\phi tanϕ+cotϕ≡2csc2ϕfor ϕ≠nπ2,n∈Z\phi \neq \frac{n\pi}{2}, n \in \mathbb{Z}ϕ=2nπ,n∈Z.
Using the identity in part (a), or otherwise, prove that
cot2ϕ−tan2ϕ≡4cot2ϕcsc2ϕ \cot^2 \phi - \tan^2 \phi \equiv 4 \cot 2\phi \csc 2\phi cot2ϕ−tan2ϕ≡4cot2ϕcsc2ϕfor ϕ≠nπ2,n∈Z\phi \neq \frac{n\pi}{2}, n \in \mathbb{Z}ϕ=2nπ,n∈Z.
Hence solve, for −π2<α<π2-\frac{\pi}{2} < \alpha < \frac{\pi}{2}−2π<α<2π,
4cot2αcsc2α=3tan2α+1 4 \cot 2\alpha \csc 2\alpha = 3 \tan^2 \alpha + 1 4cot2αcsc2α=3tan2α+1giving your answers to 2 decimal places.
317 exam-style questions on OCR (MEI) A Level Maths 1.7 Trigonometry, covering 1.7.1 Solve right-angled triangles, 1.7.2 Definitions of sin, cos and tan for any angle, 1.7.3 Graphs of sin, cos and tan, 1.7.4 Exact values of trig functions (degrees), 1.7.5 Area of a triangle, 1.7.6 Sine and cosine rules, 1.7.7 Identity tan = sin/cos, 1.7.8 Identity sin^2 + cos^2 = 1, 1.7.9 Solve simple trigonometric equations, 1.7.10 Exact values of trig functions (radians) (A-level only), 1.7.11 Inverse trigonometric functions (A-level only), 1.7.12 Radians and degree conversion (A-level only), 1.7.13 Arc length and area of a sector (A-level only), 1.7.14 Small angle approximations (A-level only), 1.7.15 Sec, cosec and cot functions (A-level only), 1.7.16 Graphs of reciprocal trig functions (A-level only), 1.7.17 Pythagorean identities for sec and cosec (A-level only), 1.7.18 Compound angle formulae (A-level only), 1.7.19 Double angle identities (A-level only), 1.7.20 Expressions for a cos θ ± b sin θ (A-level only), 1.7.21 Use identities to solve equations (A-level only), 1.7.22 Proofs involving trigonometric functions (A-level only), 1.7.23 Trig to solve problems in context (A-level only), and 1.7 Trigonometry. Each one has a worked solution and a mark scheme showing where the marks go.