8- Prove that -i sin A-sin 3 A-sin 5 A-cos A-cos 3 A-cos 5 A-tan 3 Aii ii cos 3 A -2 cos 5 A-cos 7 A -cos A -2 cos 3 A -cos 5 A -cos 5 A -cos 3 Acos 4 A -cos 3 A -cos 2 A - 3 Avx sin A-2 sin 3 A-sin 5 A-sin 3 A-2 sin 5 A-sin 7 A-sin 3 A-sin 5 Axi sin -2 sin-sin -cos -2 cos-cos -

(i) We have, LHS=sin A + sin 3A + sin 5A/cosA+cos3A+cos5A =(sin5A+sinA)+sin3A/(cos5A+cosA)+cos3A =2sin ( 5A+A/2 )cos ( 5A A/2 )+sin3A/2cos ( 5A+A/2 )cos2sin ( ...

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Byju's Answer

Standard XII

Mathematics

Sum of Trigonometric Ratios in Terms of Their Product

8. Prove that...

Question

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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Solution

(i) We have,
$L H S = \frac{s i n A + s i n 3 A + s i n 5 A}{c o s A + c o s 3 A + c o s 5 A} = \frac{(s i n 5 A + s i n A) + s i n 3 A}{(c o s 5 A + c o s A) + c o s 3 A} = \frac{2 s i n (\frac{5 A + A}{2}) c o s (\frac{5 A - A}{2}) + s i n 3 A}{2 c o s (\frac{5 A + A}{2}) c o s 2 s i n (\frac{5 A - A}{2}) + c o s 3 A} = \frac{2 s i n 3 A c o s 2 A + s i n 3 A}{2 c o s 3 A c o s 2 A + c o s 3 A} = \frac{s i n 3 A (2 c o s 2 A + 1)}{c o s 3 A (2 c p s 2 A + 1)} = \frac{s i n 3 A}{c o s 3 A} = t a n 3 A = R H S ∴ \frac{s i n A + s i n 3 A + s i n 5 A}{c o s A + c o s 3 A + c o s 5 A} = t a n 3 A$

(ii) $\frac{c o s 3 A + 2 c o s 5 A + c o s 7 A}{c o s A + 2 c o s 3 A + c o s 5 A} = \frac{c o s 5 A}{c o s 3 A} L H S = \frac{c o s 3 A + 2 c o s 5 A + c o s 7 A}{c o s A + 2 c o s 3 A + c o s 5 A} = \frac{(c o s 7 A + c o s 3 A) + 2 c o s 5 A}{(c o s 5 A + c o s A) + 2 c o s 3 A} = \frac{2 c o s (\frac{7 A + 3 A}{2}) c o s (\frac{7 A - 3 A}{2}) + 2 c o s 5 A}{2 c o s (\frac{5 A + A}{2}) c o s (\frac{5 A - A}{2}) + c o s 3 A} = \frac{2 c o s 5 A c o s 2 A + 2 c o s 5 A}{2 c o s 3 A c o s 2 A + 2 c o s 3 A} = \frac{2 c o s 5 A (c o s 2 A + 1)}{2 c o s 3 A (c o s 2 A + 1)} = \frac{2 c o s 5 A}{2 c o s 3 A} ∴ \frac{c o s 3 A + 2 c o s 5 A + c o s 7 A}{c o s A + 2 c o s 3 A + c o s 5 A} = \frac{c o s 5 A}{c o s 3 A}$

(iii) $\frac{c o s 4 A + c o s 3 A + c o s 2 A}{c o s 4 A + s i n 3 A + s i n 2 A} = c o t 3 A L H S = \frac{(c o s 4 A + c o s 2 A) + c o s 3 A}{(s i n 4 A + s i n 2 A) + s i n 3 A} = \frac{2 c o s (\frac{4 A + 2 A}{2}) c o s (\frac{4 A - 2 A}{2}) + c o s 3 A}{2 s i n (\frac{4 A + 2 A}{2}) c o s (\frac{4 A - 2 A}{2}) + s i n 3 A} = \frac{2 c o s 3 A c o s A + c o s 3 A}{2 s i n 3 A c o s A + s i n 3 A} = \frac{c o s 3 A (2 c o s A + 1)}{s i n 3 A (2 c o s A + 1)} = \frac{c o s 3 A}{s i n A} = c o t 3 A = R H S ∴ \frac{c o s 4 A + c o s 3 A + c o s 2 A}{c o s 4 A + s i n 3 A + s i n 2 A} = c o t 3 A$

(iv) $\frac{s i n 3 A + s i n 5 A + s i n 7 A + s i n 9 A}{c o s 3 A + c o s 5 A + c o s 7 A + c o s 9 A} = t a n 6 A$
We have,
$LHS=fracsin3A+sin5A+sin7A+sin9Acos3A+cos5A+cos7A+cos9A=(sin9A+sin3A)+(sin7A+sin5A)(cos9A+cos3A)+(cos7A+cos5A)=2sin(9A+3A2)cos(9A−3A2)+2sin(7A+5A2)cos(7A−5A2)2cos(9A+3A2)cos(9A−3A2)+2cos(9A+3A2)+2cos2sin(7A+5A2)cos(7A−5A2)=2sin6A cos3A+2sin6A cosA2cos6A cos3A+2cos6A cosA=2sin6A(cos3A+cosA)2cos6A(cos3A+cosA)=sin6Acos6A=tan6A=RHS∴sin3A+sin5A+sin7A+sin9Acos3A+cos5A+cos7A+cos9A=tan6A$

(v) $\frac{s i n 5 A - s i n 7 A + s i n 8 A - s i n 4 A}{c o s 4 A + c o s 7 A + c o s 7 A - c o s 5 A - c o s 8 A} = c o t 6 A$
We have,
$L H S = \frac{s i n 5 A - s i n 7 A + s i n 8 A - s i n 4 A}{c o s 4 A + c o s 7 A + c o s 7 A - c o s 5 A - c o s 8 A} = \frac{- (s i n 7 A - s i n 5 A) + (s i n 8 A - s i n 4 A)}{- (c o s 7 A - c o s 5 A) - (c o s 8 A - c o s 4 A)} = \frac{- [2 s i n (\frac{7 A - 5 A}{2}) c o s (\frac{7 A + 5 A}{2})] + [2 s i n (\frac{8 A - 4 A}{2}) c o s (\frac{8 A + 4 A}{2})]}{- 2 s i n (\frac{7 A + 5 A}{2}) s i n (\frac{7 A - 5 A}{2}) - [- 2 s i n (\frac{8 A + 4 A}{2}) s i n (\frac{8 A - 4 A}{2})]} = \frac{- 2 s i n A c o s 6 A + 2 s i n 2 A c o s 6 A}{- 2 s i n 6 A s i n A + 2 s i n 6 A s i n 2 A} = \frac{2 c o s 6 A [s i n A + s i n 2 A]}{s i n A + s i n 2 A} = \frac{c o s 6 A}{s i n 6 A} = c o t 6 A = R H S ∴ \frac{s i n 5 A - s i n 7 A + s i n 8 A - s i n 4 A}{c o s 4 A + c o s 7 A - c o s 5 A - c o s 8 A} = t a n 6 A$

(vi) $\frac{s i n 5 A + c o s 2 A - s i n 6 A c o s A}{s i n A s i n 2 A - c o s 2 A c o s 3 A} = t a n A$
We have,
$L H S = \frac{s i n 5 A + c o s 2 A - s i n 6 A c o s A}{s i n A s i n 2 A - c o s 2 A c o s 3 A} = \frac{2 (s i n 5 A c o s 2 A) + s i n 6 c o s A}{2 (s i n A s i n 2 A - c o s 2 A c o s 3 A)} = \frac{2 s i n 5 A c o s 2 A - 2 s i n 6 A c o s A}{2 s i n A s i n 2 A - 2 c o s 2 A c o s 3 A} = \frac{s i n (5 A + 2 A) + s i n (5 A - 2 A) - [s i n (6 A + A) + s i n (6 A - 2 A)]}{c o s (2 A - A) - c o s (2 A + A) - [c o s (3 A + 2 A) + c o s (3 A - 2 A)]} = \frac{s i n 7 A + s i n 3 A - s i n 7 A - s i n 5 A}{c o s A - c o s 3 A - c o s 5 A - c o s A} = \frac{s i n 3 A - s i n 5 A}{- c o s 3 A - c o s 5 A} = \frac{- (s i n 5 A - s i n 3 A)}{- (c o s 5 A + 3 A)} = \frac{s i n 5 A - s i n 3 A}{c o s 5 A + c o s 3 A} = \frac{2 s i n (\frac{5 A - 3 A}{2}) c o s (\frac{5 A + 3 A}{2})}{2 c o s (\frac{5 A + 3 A}{2}) c o s (\frac{5 A - 3 A}{2})} = \frac{s i n A c o s 4 A}{c o s 4 A c o s A} = \frac{s i n A}{c o s A} = t a n A = R H S ∴ \frac{s i n 5 A c o s 2 A - s i n 6 A c o s A}{s i n A s i n 2 A - c o s 2 A c o s 3 A} = t a n A$

(vii) $\frac{s i n 11 A + s i n A + s i n 7 A + s i n 3 A}{c o s 11 A s i n A + c o s 7 A s i n 3 A} = t a n 8 A W e h a v e, L H S = \frac{s i n 11 A s i n A + s i n 7 A s i n 3 A}{c o s 11 A s i n A + c o s 7 A s i n 3 A} = \frac{2 (s i n 11 A s i n A + s i n 7 A s i n 3 A)}{2 c o s 11 A s i n A + c o s 7 A s i n 3 A} = \frac{2 s i n 11 A 5 i n A + 2 s i n 7 A s i n 3 A}{2 c p s 11 A s i n A + 2 c o s 7 A s i n 3 A} = \frac{c o s (11 A - A) - c o s (11 A + A) + c o s (7 A - 3 A) - c o s (7 A + 3 A)}{s i n (11 A + A) - s i n (11 A - A) + s i n (7 A + 3 A) - s i n (7 A - 3 A)} = \frac{c o s 10 A - c o s 12 A + c o s 4 A - c o s 10 A}{s i n 12 A - s i n 10 A + s i n 10 A - s i n 4 A} = \frac{- (c o s 12 A - c o s 4 A)}{s i n 12 A - s i n 4 A} = \frac{[- 2 s i n (\frac{12 A + 4 A}{2}) s i n (\frac{12 A - 4 A}{2})]}{2 s i n (\frac{12 A - 4 A}{2}) c o s s i n (\frac{12 A + 4 A}{2})} = \frac{2 s i n 8 A s i n 4 A}{2 s i n 4 A c o s 8 A} = \frac{s i n 8 A}{c o s 8 A} = t a n 8 A = R H S ∴ \frac{s i n 11 A s i n A s i n 7 A s i n 3 A}{s i n 11 A s i n A + c o s 7 A s i n 3 A} = t a n 8 A$

(viii) $\frac{s i n 3 A c o s 4 A - s i n A c o s 2 A}{s i n 4 A s i n A + c o s 6 A c o s A} = t a n 2 A L H S = \frac{s i n 3 A c o s 4 A - s i n A c o s 2 A}{s i n 4 A s i n A + c o s 6 A c o s A} = \frac{2 s i n 3 A c o s 4 A - s i n A c o s 2 A}{2 (s i n 4 A s i n A + c o s 6 A c o s A)} = \frac{2 s i n 3 A c o s 4 A - 2 s i n A ‘ c o s 2 A}{2 s i n 4 A s i n A + 2 c o s 6 A c o s A} = \frac{s i n (4 A + 3 A) - s i n (4 A + - 3 A) - [s i n (2 A + A) - s i n (2 A - A)]}{c o s s i n (4 A - A) - c o s (4 A + A) + c o s (6 A + A) + c o s (6 A - A)} = \frac{s i n (7 A) - s i n (A) - s i n (3 A) + s i n (A)}{c o s (3 A) - c o s (5 A) + c o s (7 A) + c o s (5 A)} = \frac{s i n (7 A) - s i n (3 A)}{c o s (3 A) + c o s (7 A)} = \frac{2 s i n (\frac{7 A - 3 A}{2}) c o s (\frac{7 A + 3 A}{2})}{2 c o s (\frac{7 A + 3 A}{2}) c o s (\frac{7 A - 3 A}{2})} = \frac{s i n 2 A}{c o s 2 A} = t a n 2 A = R H S$

(ix) $\frac{s i n A s i n 2 A + s i n 3 A s i n 6 A}{s i n A c o s 2 A + c o s 3 A c o s 6 A} = t a n 5 A$
We have,
$L H S = \frac{s i n A s i n 2 A + s i n 3 A s i n 6 A}{s i n A c o s 2 A + c o s 3 A c o s 6 A} = \frac{2 [s i n A s i n 2 A + s i n 3 A s i n 6 A]}{2 [s i n A c o s 2 A + s i n 3 A c o s 6 A]} = \frac{2 s i n 2 A s i n A + 2 s i n 6 A s i n 3 A}{2 c o s 2 A s i n A + 2 c o s 6 A s i n 3 A} = \frac{c o s (2 A - A) - c o s (2 A + A) + c o s (6 A - 3 A) - c o s (6 A + 3 A)}{s i n (2 A + A) - s i n (2 A - A) + s i n (6 A + 3 A) - s i n (6 A - 3 A)} = \frac{c o s A - c o s 3 A - c o s 9 A}{s i n 3 A - s i n A + s i n 9 A - s i n 3 A} = \frac{c o s A - c o s 9 A}{s i 9 A - s i n A} = \frac{- [c o s 9 A - c o s A]}{s i n 9 A - s i n A} = \frac{(- 2 s i n (\frac{9 A + A}{2})) \times (\frac{9 A - A}{2})}{- 2 s i n (\frac{9 A - A}{2}) \times c o s (\frac{9 A + A}{2})} = \frac{s i n 5 A s i n 4 A}{s i n 4 A c o s 5 A} = t a n 5 A = R H S ∴ \frac{s i n A s i n 2 A + s i n 3 A s i n 6 A}{s i n A c o s 2 A + s i n 3 A + c o s 6 A} = t a n 5 A$

(x) $\frac{s i n A + 2 s i n 3 A + s i n 5 A}{s i n 3 A + 2 s i n 5 A + s i n 7 A} = \frac{s i n 3 A}{s i n 5 A}$
We have.
$L H S = \frac{s i n A + 2 s i n 3 A + s i n 5 A}{s i n 3 A + 2 s i n 5 A + s i n 7 A} = \frac{s i n 5 A + 2 s i n A + 2 s i n 3 A}{s i n 7 A + 2 s i n 3 A + 2 s i n 5 A} = \frac{2 s i n (\frac{5 A + A}{2}) c o s (\frac{5 A - A}{2}) + 2 s i n 3 A}{2 s i n (\frac{7 A + 3 A}{2}) c o s (\frac{7 A - 3 A}{2}) + 2 s i n 5 A} = \frac{2 s i n 3 A c o s 2 A + 2 s i n 3 A}{2 s i n 5 A c o s 2 A + s i n 5 A} = \frac{2 s i n 3 A (c o s 2 A + 1)}{2 s i n 5 A (c o s 2 A + 1)} = \frac{s i n 3 A}{s i n 5 A} ∴ \frac{s i n A + 2 s i n 3 A + s i n 5 A}{s i n 3 A + 2 s i n 5 A + s i n 7 A} = \frac{s i n 3 A}{s i n 5 A}$

(xi) $\frac{s i n (θ + ϕ) - 2 s i n θ + s i n (θ - ϕ)}{c o s (θ + ϕ) - 2 c o s θ + c o s (θ - ϕ)} = \frac{s i n (θ + ϕ) + s i n (θ - ϕ) - 2 s i n θ}{c o s (θ + ϕ) + c o s (θ - ϕ) - 2 c o s θ} = \frac{2 s i n [\frac{(θ + ϕ) + (θ - ϕ)}{2}] c o s [\frac{(θ + ϕ) - (θ - ϕ)}{2}] - 2 s i n θ}{2 c o s [\frac{(θ + ϕ) + (θ - ϕ)}{2}] c o s [\frac{(θ + ϕ) + (θ - ϕ)}{2}] - 2 c o s θ} = \frac{2 s i n (θ) c o s (ϕ) - 2 s i n θ}{2 c o s (θ) c o s (ϕ) - 2 c o s θ} = \frac{2 s i n θ (c o s ϕ - 1)}{2 c o s θ (c o s ϕ - 1)} = \frac{s i n θ}{c o s θ} = t a n θ ∴ \frac{s i n (θ + ϕ) - 2 s i n θ + s i n (θ - ϕ)}{c o s (θ + ϕ) - 2 c o s θ + c o s (θ - ϕ)} = t a n θ$

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