Advertisements
Advertisements
Question
Prove that:
`sin A + sin 2A + sin 4A + sin 5A = 4 cos (A/2) cos((3A)/2)sin3A`
Advertisements
Solution
Consider LHS:
\[ \sin A + \sin 2A + \sin 4A + \sin 5A\]
\[ = 2\sin \left( \frac{A + 2A}{2} \right) \cos \left( \frac{A - 2A}{2} \right) + 2\sin \left( \frac{4A + 5A}{2} \right) \cos \left( \frac{4A - 5A}{2} \right) \left\{ \because \sin A + \sin B = 2\sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \right\}\]
\[ = 2\sin \left( \frac{3}{2}A \right) \cos \left( - \frac{A}{2} \right) + 2\sin \left( \frac{9}{2}A \right) \cos \left( - \frac{A}{2} \right)\]
\[= 2\sin \left( \frac{3}{2}A \right) \cos \left( \frac{A}{2} \right) + 2\sin \left( \frac{9}{2}A \right) \cos \left( \frac{A}{2} \right)\]
\[ = 2\cos \left( \frac{A}{2} \right)\left\{ \sin \frac{3}{2}A + \sin \frac{9}{2}A \right\}\]
\[ = 2\cos \left( \frac{A}{2} \right) \times 2\sin \left( \frac{\frac{3}{2}A + \frac{9}{2}A}{2} \right) \cos \left( \frac{\frac{3}{2}A - \frac{9}{2}A}{2} \right)\]
\[ = 4\cos \left( \frac{A}{2} \right) \sin 3A \cos \left( - \frac{3}{2}A \right)\]
\[ = 4\cos \frac{A}{2} \cos \left( \frac{3A}{2} \right) \sin 3A\]
= RHS
Hence, LHS = RHS
APPEARS IN
RELATED QUESTIONS
Prove that:
Prove that:
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
Prove that:
cos 40° cos 80° cos 160° = \[- \frac{1}{8}\]
Prove that:
sin 20° sin 40° sin 80° = \[\frac{\sqrt{3}}{8}\]
Prove that tan 20° tan 30° tan 40° tan 80° = 1.
Prove that:
sin 20° sin 40° sin 60° sin 80° = \[\frac{3}{16}\]
Show that:
sin A sin (B − C) + sin B sin (C − A) + sin C sin (A − B) = 0
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 12x + sin 4x
Prove that:
sin 50° + sin 10° = cos 20°
Prove that:
cos 20° + cos 100° + cos 140° = 0
Prove that:
Prove that:
sin 47° + cos 77° = cos 17°
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:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
If \[x \cos\theta = y \cos\left( \theta + \frac{2\pi}{3} \right) = z \cos\left( \theta + \frac{4\pi}{3} \right)\], prove that \[xy + yz + zx = 0\]
If (cos α + cos β)2 + (sin α + sin β)2 = \[\lambda \cos^2 \left( \frac{\alpha - \beta}{2} \right)\], write the value of λ.
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
If sin 2A = λ sin 2B, then write the value of \[\frac{\lambda + 1}{\lambda - 1}\]
cos 40° + cos 80° + cos 160° + cos 240° =
If A, B, C are in A.P., then \[\frac{\sin A - \sin C}{\cos C - \cos A}\]=
If sin (B + C − A), sin (C + A − B), sin (A + B − C) are in A.P., then cot A, cot B and cot Care in
Express the following as the sum or difference of sine or cosine:
cos(60° + A) sin(120° + A)
Prove that:
cos 20° cos 40° cos 80° = `1/8`
Prove that:
tan 20° tan 40° tan 80° = `sqrt3`.
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`.
If tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:
