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Implicit Differentiation

47. Prove tha...

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47. Prove that cosθ .cos(θ /2) - cos3θ .cos(9θ /2) = sin7θ .sin8θ

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(cos θ + sin θ)^{2} + (cos θ - sin θ)^{2} = 2

\frac{cos 5 x + cos 4 x}{2 cos 3 x - 1} = cos x + cos 2 x

cos 2 x cos \frac{x}{2} - cos 3 x cos \frac{9 x}{2} = sin 5 x sin \frac{5 x}{2}

cos θ + sec θ = - 2, θ \in [0, 2 π],

{sin}^{8} θ + {cos}^{8} θ

s i n x + s i n y = 2 s i n \frac{x + y}{2} c o s \frac{x - y}{2}

and hence show that

\frac{s i n x + s i n 3 x}{c o s x + c o s 3 x} = t a n 2 x

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