Advertisements
Advertisements
प्रश्न
Prove that:
2 cos `pi/13` cos \[\frac{9\pi}{13} + \text{cos} \frac{3\pi}{13} + \text{cos} \frac{5\pi}{13}\] = 0
Advertisements
उत्तर
LHS = 2 cos `pi/13` cos \[\frac{9\pi}{13} + \text{cos} \frac{3\pi}{13} + \text{cos} \frac{5\pi}{13}\]
`= 2 cos pi/13 cos (9pi)/13 + 2(cos ((3pi)/13 + (5pi)/13)/2) xx (cos ((3pi)/13 - (5pi)/13)/2)`
`[∵ cos "C" + cos "D" = 2 cos (("C + D")/2) cos (("C - D")/2)]`
`= 2 cos pi/13 cos (9pi)/13 + 2(cos ((8pi)/13)/2) xx (cos (-(2pi)/13)/2)`
`= 2 cos pi/13 cos (9pi)/13 + 2cos (4pi)/13 cos ((-pi)/13)`
[∵ cos(-θ) = cos θ]
`= 2 cos pi/13 cos (9pi)/13 + 2cos (4pi)/13 cos pi/13`
`= 2 cos pi/13 (cos (9pi)/13 + cos (4pi)/13)`
[take 2 cos `pi/3` as common]
`= 2 cos pi/13 (2 cos (((9pi + 4pi)/13))/2 cos ((9pi - 4pi)/13)/2)`
`= 2 cos pi/13 (2 cos (13pi)/(13 xx 2) cos (5pi)/(13 xx 2))`
`= 2 cos pi/13 (2 cos pi/2 cos (5pi)/2)`
`= 2 cos pi/13 (0 xx cos (5pi)/(13 xx 2))`
= 0 = RHS
Hence proved.
APPEARS IN
संबंधित प्रश्न
Express each of the following as the product of sines and cosines:
sin 5x − sin x
Prove that:
Prove that:
If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ
If A + B = \[\frac{\pi}{3}\] and cos A + cos B = 1, then find the value of cos \[\frac{A - B}{2}\].
Write the value of \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]
If sin x + sin y = \[\sqrt{3}\] (cos y − cos x), then sin 3x + sin 3y =
Express the following as the sum or difference of sine or cosine:
`sin "A"/8 sin (3"A")/8`
Prove that:
cos 20° cos 40° cos 80° = `1/8`
Prove that cos 20° cos 40° cos 60° cos 80° = `3/16`.
