Advertisements
Advertisements
प्रश्न
Prove that:
cos 40° cos 80° cos 160° = \[- \frac{1}{8}\]
Advertisements
उत्तर
\[ = \frac{1}{2}\left[ 2\cos 40^\circ \cos 80^\circ \right] \cos 160^\circ\]
\[ = \frac{1}{2}\left[ \cos \left( 40^\circ + 80^\circ \right) + \cos \left( 40^\circ - 80^\circ \right) \right] \cos 160^\circ\]
\[ = \frac{1}{2}\left[ \cos 120^\circ + \cos \left( - 40^\circ \right) \right] \cos 160^\circ\]
\[ = \frac{1}{2}\cos \left( 160^\circ \right)\left[ - \frac{1}{2} + \cos 40^\circ \right]\]
\[ = - \frac{1}{4}\cos 160^\circ + \frac{1}{2}\cos 160^\circ \cos 40^\circ\]
\[= - \frac{1}{4}\cos 160^\circ + \frac{1}{4}\left[ 2\cos 160^\circ \cos 40^\circ \right]\]
\[ = - \frac{1}{4}\cos 160^\circ + \frac{1}{4}\left[ \cos \left( 160^\circ + 40^\circ \right) + \cos \left( 160^\circ - 40^\circ \right) \right]\]
\[ = - \frac{1}{4}\cos 160^\circ + \frac{1}{4}\left[ \cos 200^\circ + \cos 120^\circ \right]\]
\[ = - \frac{1}{4}\cos 160^\circ + \frac{1}{4}\left[ \cos \left( 360^\circ - 160^\circ \right) - \frac{1}{2} \right]\]
\[ = - \frac{1}{4}\cos 160^\circ + \frac{1}{4}\cos 160^\circ - \frac{1}{8} \left[ \because \cos \left( 360^\circ - 160^\circ \right) = \cos 160^\circ \right]\]
\[ = - \frac{1}{8} = RHS\]
APPEARS IN
संबंधित प्रश्न
Prove that:
Prove that:
sin 20° sin 40° sin 80° = \[\frac{\sqrt{3}}{8}\]
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
Show that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
Prove that:
sin 50° + sin 10° = cos 20°
Prove that:
sin 23° + sin 37° = cos 7°
Prove that:
Prove that:
cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A
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:
If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ
Write the value of sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
If cos A = m cos B, then write the value of \[\cot\frac{A + B}{2} \cot\frac{A - B}{2}\].
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° =
cos 35° + cos 85° + cos 155° =
If A, B, C are in A.P., then \[\frac{\sin A - \sin C}{\cos C - \cos A}\]=
Express the following as the sum or difference of sine or cosine:
cos(60° + A) sin(120° + A)
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 sum or difference of sine or cosine:
cos 7θ sin 3θ
Express the following as the product of sine and cosine.
sin A + sin 2A
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
If sin(y + z – x), sin(z + x – y), sin(x + y – z) are in A.P, then prove that tan x, tan y and tan z are in A.P.
If cosec A + sec A = cosec B + sec B prove that cot`(("A + B"))/2` = tan A tan B.
If secx cos5x + 1 = 0, where 0 < x ≤ `pi/2`, then find the value of x.
