Prove that following identities-cos 3-sin 3 -sin 3-cos 3 -3-4sin 4 -

Cos^3 θ sin 3 θ + sin^3 θ cos 3 θ=3/4sin 4 θ LHS = cos^3 θ sin 3 θ + sin^3 θ cos 3 θ = (cos 3 θ + 3 cos θ/4 )sin 3 θ+ (3 sin θ sin 3 θ/4 )cos 3 θ ∵ sin 3 θ = 3 ...

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Standard X

Mathematics

Trigonometric Identities

Prove that fo...

Question

Prove that following identities:

$c o s^{3} θ s i n 3 θ + s i n^{3} θ c o s 3 θ = \frac{3}{4} s i n 4 θ$

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Solution

$c o s^{3} θ s i n 3 θ + s i n^{3} θ c o s 3 θ = \frac{3}{4} s i n 4 θ$

LHS = $c o s^{3} θ s i n 3 θ + s i n^{3} θ c o s 3 θ$

$= (\frac{c o s 3 θ + 3 c o s θ}{4}) s i n 3 θ + (\frac{3 s i n θ - s i n 3 θ}{4}) c o s 3 θ {\begin{matrix} ∵ s i n 3 θ = 3 s i n θ - 4 s i n^{3} θ c o s 3 θ = 4 c o s^{3} θ - 3 c o s θ \end{matrix}} = \frac{1}{4} [3 (s i n 3 θ c o s θ + s i n θ c o s 3 θ) + c o s 3 θ s i n 3 θ - s i n 3 θ c o s 3 θ] = \frac{1}{4} [3 s i n (3 θ + θ) + 0] = \frac{3}{4} s i n 4 θ$

So,

$c o s^{3} θ s i n 3 θ + s i n^{3} θ c o s 3 θ = \frac{3}{4} s i n 4 θ$

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Similar questions

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Q. Prove the following trigonometric identities.

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Prove that following identities: cos3 θ sin 3θ+sin3 θ cos 3θ=34sin 4 θ

Prove that following identities:

$c o s^{3} θ s i n 3 θ + s i n^{3} θ c o s 3 θ = \frac{3}{4} s i n 4 θ$