Advertisements
Advertisements
Question
Prove that:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]
Advertisements
Solution
\[LHS = \cos 20^\circ \cos 40^\circ \cos 80^\circ\]
\[ = \frac{1}{2}\left[ 2\cos 20^\circ \cos 40^\circ \right] \cos 80^\circ\]
\[ = \frac{1}{2}\left[ \cos \left( 20^\circ + 40^\circ \right) + \cos\left( 20^\circ - 40^\circ \right) \right] \cos 80^\circ\]
\[ = \frac{1}{2}\left[ \cos 60^\circ + \cos \left( - 20^\circ \right) \right] \cos 80^\circ\]
\[ = \frac{1}{2}\cos 80^\circ\left[ \frac{1}{2} + \cos 20^\circ \right]\]
\[ = \frac{1}{4}cos 80^\circ + \frac{1}{2}\cos 80^\circ \cos 20^\circ\]
\[= \frac{1}{4}\cos 80^\circ + \frac{1}{4}\left[ 2\cos 80^\circ \cos 20^\circ \right]\]
\[ = \frac{1}{4}\cos 80^\circ + \frac{1}{4}\left[ \cos \left( 80^\circ + 20^\circ \right) + \cos \left( 80^\circ - 20^\circ \right) \right]\]
\[ = \frac{1}{4}\cos 80^\circ + \frac{1}{4}\left[ \cos 100^\circ + \cos 60^\circ \right]\]
\[ = \frac{1}{4}\cos 80^\circ + \frac{1}{4}\left[ \cos \left( 180^\circ - 80^\circ \right) + \frac{1}{2} \right]\]
\[ = \frac{1}{4}\cos 80^\circ - \frac{1}{4}\cos 80^\circ + \frac{1}{8} \left\{ \because \cos \left( 180^\circ - 80^\circ \right) = - \cos 80^\circ \right\}\]
\[ = \frac{1}{8} = RHS\]
APPEARS IN
RELATED QUESTIONS
Prove that:
Prove that:
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
Prove that:
sin 20° sin 40° sin 60° sin 80° = \[\frac{3}{16}\]
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 5x − sin x
Express each of the following as the product of sines and cosines:
sin 2x + cos 4x
Prove that:
cos 100° + cos 20° = cos 40°
Prove that:
sin 50° + sin 10° = cos 20°
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:
cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
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 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 \[\left( \frac{A + B}{2} \right)\].
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
If A + B = \[\frac{\pi}{3}\] and cos A + cos B = 1, then find the value of cos \[\frac{A - B}{2}\].
Write the value of \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]
If sin 2 θ + sin 2 ϕ = \[\frac{1}{2}\] and cos 2 θ + cos 2 ϕ = \[\frac{3}{2}\], then cos2 (θ − ϕ) =
The value of sin 78° − sin 66° − sin 42° + sin 60° is ______.
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:
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 7"A" +cos 5"A")/(sin 7"A" −sin 5"A")` = cot A
If cos A + cos B = `1/2` and sin A + sin B = `1/4`, prove that tan `(("A + B")/2) = 1/2`
If tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:
