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:
Show that :
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 12x + sin 4x
Prove that:
cos 100° + cos 20° = cos 40°
Prove that:
sin 50° + sin 10° = cos 20°
Prove that:
sin 40° + sin 20° = cos 10°
Prove that:
Prove that:
Prove that:
Prove that:
sin 47° + cos 77° = cos 17°
Prove that:
cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A
Prove that:
cos A + cos 3A + cos 5A + cos 7A = 4 cos A cos 2A cos 4A
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:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), prove that tan A tan B tan C + tan 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 \[xy + yz + zx = 0\]
Write the value of sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
If sin A + sin B = α and cos A + cos B = β, then write the value of tan \[\left( \frac{A + B}{2} \right)\].
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.
cos 40° + cos 80° + cos 160° + cos 240° =
sin 47° + sin 61° − sin 11° − sin 25° is equal to
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 (7"A")/3 sin (5"A")/3`
Express the following as the product of sine and cosine.
sin 6θ – sin 2θ
Prove that:
tan 20° tan 40° tan 80° = `sqrt3`.
Prove that:
(cos α – cos β)2 + (sin α – sin β)2 = 4 sin2 `((alpha - beta)/2)`
Prove that:
sin A sin(60° + A) sin(60° – A) = `1/4` sin 3A
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`
Prove that:
`(cos 7"A" +cos 5"A")/(sin 7"A" −sin 5"A")` = cot A
Evaluate-
cos 20° + cos 100° + cos 140°
Evaluate:
sin 50° – sin 70° + sin 10°
