Advertisements
Advertisements
Question
Prove that:
cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A
Advertisements
Solution
Consider LHS:
\[ \cos 3A + cos 5A + \cos 7A + \cos 15A\]
\[ = 2\cos \left( \frac{3A + 5A}{2} \right) \cos \left( \frac{3A - 5A}{2} \right) + 2\cos \left( \frac{7A + 15A}{2} \right) \cos \left( \frac{7A - 15A}{2} \right) \left\{ \because \cos A + \cos B = 2\cos \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \right\}\]
\[ = 2\cos 4A \cos\left( - A \right) + 2\cos 11A \cos\left( - 4A \right)\]
\[= 2\cos 4A \cos A + 2\cos 11A \cos 4A\]
\[ = 2\cos 4A \left\{ \cos A + \cos 11A \right\}\]
\[ = 2\cos 4A \times \left\{ 2\cos \left( \frac{A + 11A}{2} \right) \cos \left( \frac{A - 11A}{2} \right) \right\}\]
\[ = 4\cos 4A \cos 6A \cos\left( - 5A \right)\]
\[ = 4\cos 4A \cos 5A \cos 6A\]
= RHS
Hence, LHS = RHS
APPEARS IN
RELATED QUESTIONS
Show that :
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
Prove that:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]
Prove that
\[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x\]
If α + β = \[\frac{\pi}{2}\], show that the maximum value of cos α cos β is \[\frac{1}{2}\].
Express each of the following as the product of sines and cosines:
sin 2x + cos 4x
Prove that:
sin 38° + sin 22° = sin 82°
Prove that:
cos 80° + cos 40° − cos 20° = 0
Prove that:
cos 20° + cos 100° + cos 140° = 0
Prove that:
Prove that:
Prove that:
sin 47° + cos 77° = cos 17°
Prove that:
`sin A + sin 2A + sin 4A + sin 5A = 4 cos (A/2) cos((3A)/2)sin3A`
Prove that:
Prove that:
Prove that:
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 \[x \cos\theta = y \cos\left( \theta + \frac{2\pi}{3} \right) = z \cos\left( \theta + \frac{4\pi}{3} \right)\], prove that \[xy + yz + zx = 0\]
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
Write the value of \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
The value of sin 50° − sin 70° + sin 10° is equal to
If A, B, C are in A.P., then \[\frac{\sin A - \sin C}{\cos C - \cos A}\]=
Express the following as the sum or difference of sine or cosine:
`sin "A"/8 sin (3"A")/8`
Prove that:
tan 20° tan 40° tan 80° = `sqrt3`.
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 2"A" - cos 3"A")/(sin "2A" + sin "3A") = tan "A"/2`
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 tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:
