Prove that following identities:

$s i n 5 A = 5 c o s^{4} A s i n A - 10 c o s^{2} A s i n^{3} A + s i n^{5} A$

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Solution

We have to prove that

$s i n 5 A = 5 c o s^{4} A s i n A - 10 c o s^{2} A s i n^{3} A + s i n^{5} A$

L.H.S. = sin 5A = sin (3A + 2A)

= sin 3A cos 2A+cos 3A. sin 2A

$= (3 s i n A - 4 s i n^{3} A) (2 c o s^{2} A - 1) + (4 c o s^{3} A - 3 c o s A) 2 s i n A c o s A . = - 3 s i n A + 4 s i n^{3} A + 6 s i n A c o s^{2} A - 8 s i n^{3} A c o s^{2} A + 8 c o s^{4} A s i n A - 6 c o s^{2} A s i n A = 8 c o s^{4} A s i n A - 8 s i n^{3} A c o s^{2} A - 3 s o n A + 4 s i n^{3} A = 5 c o s^{4} A s i n A - 10 s i n^{3} A c o s^{2} A - 3 s i n A + 3 c o s^{4} A + 4 s i n^{3} A + 2 s i n^{3} A c o s^{2} A = 5 c o s^{4} A s i n A - 10 s i n^{3} A c o s^{2} A - 3 s i n A (1 - c o s^{2} A) + 2 s i n^{3} A (2 + c o s^{2} A) = 5 c o s^{4} A s i n A - 10 s i n^{3} A c o s^{2} A - 3 s i n^{3} A (1 + c o s^{2} A) + 2 s i n^{3} A (2 + c o s^{2} A)$

$= 5 c o s^{4} A s i n A - 10 s i n^{3} A c o s^{2} A - s i n^{3} A [3 (1 + c o s^{2} A) - 2 (2 + c o s^{2} A)] = 5 c o s^{4} A s i n A - 10 s i n^{3} A c o s^{2} A - s i n^{3} A [3 + 3 c o s^{2} A - 4 - 2 c o s^{2} A] = 5 c o s^{4} A s i n A - 10 s i n^{3} A c o s^{2} A - s i n^{3} A [c o s^{2} A - 1] = 5 c o s^{4} A s i n A - 10 s i n^{3} A c o s^{2} A + s i n^{5} A = R H S$

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Similar questions

\sin 5 A = 5 \cos^{4} A \sin A - 10 \sin^{3} A \cos^{2} A + \sin^{5} A

Prove that sin A - sin 3A + sin 5A - sin 7A/ sin A - sin 3A - sin 5A +sin 7A = cot 2A.

Q. Prove that

\frac{sin A + sin 5 A + sin 9 A}{cos A + cos 5 A + cos 9 A} = tan 5 A

Q. Prove -

\frac{sin A + sin 3 A + sin 5 A + sin 7 A}{cos θ + cos 3 θ + cos 5 θ + cos 7 θ} = tan 4 A

8. Prove that :
(i) $\frac{s i n A + s i n 3 A + s i n 5 A}{c o s A + c o s 3 A + c o s 5 A} = t a n 3 A$
(ii) $(i i) \frac{c o s 3 A + 2 c o s 5 A + c o s 7 A}{c o s A + 2 c o s 3 A + c o s 5 A} = \frac{c o s 5 A}{c o s 3 A}$
(iii) $\frac{c o s 4 A + c o s 3 A + c o s 2 A}{c o s 4 A + s i n 3 A + s i n 2 A} = c o t 3 A$
(iv) $\frac{s i n 3 A + s i n 5 A + s i n 7 A + s i n 9 A}{c o s 3 A + c o s 5 A + c o s 7 A + c o s 9 A} = t a n 6 A$
(v) $\frac{s i n 5 A - s i n 7 A + s i n 8 A - s i n 4 A}{c o s 4 A + c o s 7 A + c o s 7 A - c o s 5 A - c o s 8 A} = c o t 6 A$
(vi) $\frac{s i n 5 A + c o s 2 A - s i n 6 A c o s A}{s i n A s i n 2 A - c o s 2 A c o s 3 A} = t a n A$
(vii) $\frac{s i n 11 A + s i n A + s i n 7 A + s i n 3 A}{c o s 11 A s i n A + c o s 7 A s i n 3 A} = t a n 8 A$
(viii) $\frac{s i n 3 A c o s 4 A - s i n A c o s 2 A}{s i n 4 A s i n A + c o s 6 A c o s A} = t a n 2 A$
(ix) $\frac{s i n A s i n 2 A + s i n 3 A s i n 6 A}{s i n A c o s 2 A + c o s 3 A c o s 6 A} = t a n 5 A$
(x) $\frac{s i n A + 2 s i n 3 A + s i n 5 A}{s i n 3 A + 2 s i n 5 A + s i n 7 A} = \frac{s i n 3 A}{s i n 5 A}$
(xi) $\frac{s i n (θ + ϕ) - 2 s i n θ + s i n (θ + ϕ)}{c o s (θ + ϕ) - 2 c o s θ + c o s (θ + ϕ)}$

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Prove that following identities: sin 5A=5 cos4 A sin A−10 cos2 A sin3 A+sin5A

Prove that following identities:

$s i n 5 A = 5 c o s^{4} A s i n A - 10 c o s^{2} A s i n^{3} A + s i n^{5} A$