Advertisements
Advertisements
Question
Advertisements
Solution
\[\text{ LHS }= 4\cos x \cos \left( \frac{\pi}{3} + x \right) \cos \left( \frac{\pi}{3} - x \right)\]
\[ = 2\cos x\left[ 2 \cos \left( \frac{\pi}{3} + x \right) \cos \left( \frac{\pi}{3} - x \right) \right]\]
\[ = 2\cos x\left[ \cos \left( \frac{\pi}{3} + x + \frac{\pi}{3} - x \right) + \cos \left( \frac{\pi}{3} + x - \frac{\pi}{3} + 2x \right) \right] \left[ \because 2\cos A \cos B = \cos (A + B) + \cos (A - B) \right]\]
\[ = 2\cos x\left[ \cos \frac{2\pi}{3} + \cos 2x \right]\]
\[ = 2\cos x\left[ - \frac{1}{2} + \cos 2x \right]\]
\[ = - \cos x + 2\cos x \cos 2x\]
\[ = - \cos x + \cos \left( x + 2x \right) + \cos \left( x - 2x \right)\]
\[ = - \cos x + \cos 3x + \cos\left( - x \right)\]
\[ = - \cos x + \cos 3x + \cos x\]
\[ = \cos 3x\]
\[\text{ RHS }= \cos 3x\]
Hence, LHS = RHS
APPEARS IN
RELATED QUESTIONS
Prove that:
Show 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:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]
Prove that:
sin 20° sin 40° sin 60° sin 80° = \[\frac{3}{16}\]
Prove that:
\[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x\]
Prove that:
sin 50° + sin 10° = cos 20°
Prove that:
sin 23° + sin 37° = cos 7°
Prove that:
sin 105° + cos 105° = cos 45°
Prove that:
sin 40° + sin 20° = cos 10°
Prove that:
cos 55° + cos 65° + cos 175° = 0
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
\[\sin\frac{5\pi}{18} - \cos\frac{4\pi}{9} = \sqrt{3} \sin\frac{\pi}{9}\]
Prove that:
Prove that:
`sin A + sin 2A + sin 4A + sin 5A = 4 cos (A/2) cos((3A)/2)sin3A`
Prove that:
cos 20° cos 100° + cos 100° cos 140° − 140° cos 200° = −\[\frac{3}{4}\]
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 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 \[xy + yz + zx = 0\]
If cos A = m cos B, then write the value of \[\cot\frac{A + B}{2} \cot\frac{A - B}{2}\].
If sin 2A = λ sin 2B, then write the value of \[\frac{\lambda + 1}{\lambda - 1}\]
If sin α + sin β = a and cos α − cos β = b, then tan \[\frac{\alpha - \beta}{2}\]=
Express the following as the sum or difference of sine or cosine:
`sin "A"/8 sin (3"A")/8`
Express the following as the sum or difference of sine or cosine:
cos(60° + A) sin(120° + A)
Express the following as the product of sine and cosine.
sin 6θ – sin 2θ
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
Evaluate-
cos 20° + cos 100° + cos 140°
Find the value of tan22°30′. `["Hint:" "Let" θ = 45°, "use" tan theta/2 = (sin theta/2)/(cos theta/2) = (2sin theta/2 cos theta/2)/(2cos^2 theta/2) = sintheta/(1 + costheta)]`
If secx cos5x + 1 = 0, where 0 < x ≤ `pi/2`, then find the value of x.
