Advertisements
Advertisements
Question
Prove that:
Advertisements
Solution
Consider LHS:
\[ \frac{\sin 5A - \sin 7A + \sin 8A - \sin 4A}{\cos 4A + \cos 7A - \cos 5A - \cos 8A}\]
\[ = \frac{\sin 5A - \sin 7A + \sin 8A - \sin 4A}{\cos 4A - \cos 8A + \cos 7A - \sin 5A}\]
\[ = \frac{2\sin \left( \frac{5A - 7A}{2} \right) \cos \left( \frac{5A + 7A}{2} \right) + 2\sin \left( \frac{8A - 4A}{2} \right) \cos \left( \frac{8A + 4A}{2} \right)}{- 2\sin \left( \frac{4A + 8A}{2} \right) \sin \left( \frac{4A - 8A}{2} \right) - 2\sin \left( \frac{7A + 5A}{2} \right) \sin \left( \frac{7A - 5A}{2} \right)}\]
\[ \]
\[ = \frac{2\sin \left( - A \right) \cos 6A + 2\sin 2A \cos 6A}{- 2\sin 6A \sin \left( - 2A \right) - 2\sin 6A \sin A}\]
\[ = \frac{- 2\sin A \cos 6A + 2\sin 2A cos 6A}{2\sin 6A \sin 2A - 2\sin 6A \sin A}\]
\[ = \frac{2\cos 6A\left[ \sin 2A - \sin A \right]}{2\sin 6A\left[ \sin 2A - \sin A \right]}\]
\[ = \cot 6A\]
= RHS
Hence, LHS = RHS.
APPEARS IN
RELATED QUESTIONS
Prove that:
Show that :
Show that :
Prove that:
sin 20° sin 40° sin 60° sin 80° = \[\frac{3}{16}\]
Express each of the following as the product of sines and cosines:
sin 12x + sin 4x
Express each of the following as the product of sines and cosines:
cos 12x - cos 4x
Prove that:
sin 50° + sin 10° = cos 20°
Prove that:
sin 105° + cos 105° = cos 45°
Prove that:
cos 55° + cos 65° + cos 175° = 0
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
cos 80° + cos 40° − cos 20° = 0
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (− A + B + C) = 4 cos A cos Bcos C
If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ
Write the value of sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
Write the value of \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.
The value of sin 50° − sin 70° + sin 10° is equal to
If sin x + sin y = \[\sqrt{3}\] (cos y − cos x), then sin 3x + sin 3y =
If \[\tan\alpha = \frac{x}{x + 1}\] and
Express the following as the sum or difference of sine or cosine:
cos 7θ sin 3θ
Express the following as the product of sine and cosine.
cos 2A + cos 4A
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that:
sin A sin(60° + A) sin(60° – A) = `1/4` sin 3A
Prove that:
sin (A – B) sin C + sin (B – C) sin A + sin(C – A) sin B = 0
Prove that:
2 cos `pi/13` cos \[\frac{9\pi}{13} + \text{cos} \frac{3\pi}{13} + \text{cos} \frac{5\pi}{13}\] = 0
Prove that:
`(cos 7"A" +cos 5"A")/(sin 7"A" −sin 5"A")` = cot A
Evaluate-
cos 20° + cos 100° + cos 140°
If sin(y + z – x), sin(z + x – y), sin(x + y – z) are in A.P, then prove that tan x, tan y and tan z are in A.P.
If secx cos5x + 1 = 0, where 0 < x ≤ `pi/2`, then find the value of x.
