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
Prove that:
Prove that tan 20° tan 30° tan 40° tan 80° = 1.
Show that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
Express each of the following as the product of sines and cosines:
sin 5x − sin x
Express each of the following as the product of sines and cosines:
cos 12x - cos 4x
Prove that:
sin 38° + sin 22° = sin 82°
Prove that:
sin 105° + cos 105° = cos 45°
Prove that:
sin 40° + sin 20° = cos 10°
Prove that:
Prove that:
Prove that:
Prove that:
`sin A + sin 2A + sin 4A + sin 5A = 4 cos (A/2) cos((3A)/2)sin3A`
Prove that:
sin 3A + sin 2A − sin A = 4 sin A cos \[\frac{A}{2}\] \[\frac{3A}{2}\]
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
If y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ.
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 sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
Write the value of \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
If sin α + sin β = a and cos α − cos β = b, then tan \[\frac{\alpha - \beta}{2}\]=
If cos A = m cos B, then \[\cot\frac{A + B}{2} \cot\frac{B - A}{2}\]=
Prove that:
cos 20° cos 40° cos 80° = `1/8`
Prove that:
(cos α – cos β)2 + (sin α – sin β)2 = 4 sin2 `((alpha - beta)/2)`
Prove that:
`(cos 7"A" +cos 5"A")/(sin 7"A" −sin 5"A")` = cot A
Prove that cos 20° cos 40° cos 60° cos 80° = `3/16`.
Evaluate:
sin 50° – sin 70° + sin 10°
If tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:
