Advertisements
Advertisements
प्रश्न
Prove that:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]
Advertisements
उत्तर
\[LHS = \cos 20^\circ \cos 40^\circ \cos 80^\circ\]
\[ = \frac{1}{2}\left[ 2\cos 20^\circ \cos 40^\circ \right] \cos 80^\circ\]
\[ = \frac{1}{2}\left[ \cos \left( 20^\circ + 40^\circ \right) + \cos\left( 20^\circ - 40^\circ \right) \right] \cos 80^\circ\]
\[ = \frac{1}{2}\left[ \cos 60^\circ + \cos \left( - 20^\circ \right) \right] \cos 80^\circ\]
\[ = \frac{1}{2}\cos 80^\circ\left[ \frac{1}{2} + \cos 20^\circ \right]\]
\[ = \frac{1}{4}cos 80^\circ + \frac{1}{2}\cos 80^\circ \cos 20^\circ\]
\[= \frac{1}{4}\cos 80^\circ + \frac{1}{4}\left[ 2\cos 80^\circ \cos 20^\circ \right]\]
\[ = \frac{1}{4}\cos 80^\circ + \frac{1}{4}\left[ \cos \left( 80^\circ + 20^\circ \right) + \cos \left( 80^\circ - 20^\circ \right) \right]\]
\[ = \frac{1}{4}\cos 80^\circ + \frac{1}{4}\left[ \cos 100^\circ + \cos 60^\circ \right]\]
\[ = \frac{1}{4}\cos 80^\circ + \frac{1}{4}\left[ \cos \left( 180^\circ - 80^\circ \right) + \frac{1}{2} \right]\]
\[ = \frac{1}{4}\cos 80^\circ - \frac{1}{4}\cos 80^\circ + \frac{1}{8} \left\{ \because \cos \left( 180^\circ - 80^\circ \right) = - \cos 80^\circ \right\}\]
\[ = \frac{1}{8} = RHS\]
APPEARS IN
संबंधित प्रश्न
Prove that:
Prove that:
Prove that:
cos 40° cos 80° cos 160° = \[- \frac{1}{8}\]
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
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:
cos 12x + cos 8x
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:
sin 40° + sin 20° = cos 10°
Prove that:
cos 80° + cos 40° − cos 20° = 0
Prove that:
Prove that:
Prove that:
sin 3A + sin 2A − sin A = 4 sin A cos \[\frac{A}{2}\] \[\frac{3A}{2}\]
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:
If cosec A + sec A = cosec B + sec B, prove that tan A tan B = \[\cot\frac{A + B}{2}\].
Prove that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 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.
If cos A = m cos B, then write the value of \[\cot\frac{A + B}{2} \cot\frac{A - B}{2}\].
Write the value of \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.
If sin α + sin β = a and cos α − cos β = b, then tan \[\frac{\alpha - \beta}{2}\]=
cos 35° + cos 85° + cos 155° =
sin 47° + sin 61° − sin 11° − sin 25° is equal to
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 product of sine and cosine.
cos 2θ – cos θ
Prove that:
tan 20° tan 40° tan 80° = `sqrt3`.
Prove that:
sin A sin(60° + A) sin(60° – A) = `1/4` sin 3A
Evaluate:
sin 50° – sin 70° + sin 10°
If secx cos5x + 1 = 0, where 0 < x ≤ `pi/2`, then find the value of x.
