Advertisements
Advertisements
Question
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
Advertisements
Solution
\[LHS = \cos 10^\circ \cos 30^\circ \cos 50^\circ \cos 70^\circ\]
\[ = \frac{1}{2} \left[ 2\cos 10^\circ \cos 50^\circ \right] \cos 30^\circ \cos 70^\circ\]
\[ = \frac{1}{2} \left[ \cos \left( 10^\circ + 50^\circ\right) + \cos \left( 10^\circ - 50^\circ \right) \right] \cos 30^\circ \cos 70^\circ \left\{ \because 2\cos A \cos B = \cos\left( A + B \right) - \cos \left( A - B \right) \right\}\]
\[ = \frac{1}{2} \left[ \cos 60^\circ + \cos \left( - 40^\circ \right) \right] \cos 30^\circ \cos 70^\circ\]
\[ = \frac{1}{2} \left[ \frac{1}{2} + \cos 40^\circ \right]\left( \frac{\sqrt{3}}{2} \right) \times \cos 70^\circ\]
\[= \frac{\sqrt{3}}{4}\cos 70^\circ\left[ \frac{1}{2} + \cos 40^\circ \right]\]
\[ = \frac{\sqrt{3}}{8}\cos 70^\circ + \frac{\sqrt{3}}{4}\left[ \cos 70^\circ \cos 40^\circ \right]\]
\[ = \frac{\sqrt{3}}{8}\cos 70^\circ + \frac{\sqrt{3}}{8}\left[ 2\cos 70^\circ \cos 40^\circ \right]\]
\[ = \frac{\sqrt{3}}{8}\cos 70^\circ + \frac{\sqrt{3}}{8}\left[ \cos \left( 70^\circ + 40^\circ \right) + \cos \left( 70^\circ - 40^\circ \right) \right]\]
\[ = \frac{\sqrt{3}}{8}\cos 70^\circ + \frac{\sqrt{3}}{8}\left[ \cos 110^\circ + \cos 30^\circ \right]\]
\[ = \frac{\sqrt{3}}{8}\cos 70^\circ + \frac{\sqrt{3}}{8}\left[ \cos \left( 180^\circ - 70^\circ \right) + \frac{\sqrt{3}}{2} \right]\]
\[ = \frac{\sqrt{3}}{2}\cos 70^\circ - \frac{\sqrt{3}}{8}\cos 70^\circ + \frac{3}{16} \left[ \because \cos \left( 180^\circ - 70^\circ \right) = - \cos 70^\circ \right]\]
\[ = \frac{3}{16} = RHS\]
APPEARS IN
RELATED QUESTIONS
Prove that:
Prove that:
cos 40° cos 80° cos 160° = \[- \frac{1}{8}\]
Prove that:
sin 20° sin 40° sin 80° = \[\frac{\sqrt{3}}{8}\]
Prove that:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
If α + β = \[\frac{\pi}{2}\], show that the maximum value of cos α cos β is \[\frac{1}{2}\].
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:
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:
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 (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 \[\cot\frac{A + B}{2} \cot\frac{A - B}{2}\].
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 \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
Write the value of \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]
If sin α + sin β = a and cos α − cos β = b, then tan \[\frac{\alpha - \beta}{2}\]=
If cos A = m cos B, then \[\cot\frac{A + B}{2} \cot\frac{B - A}{2}\]=
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 sum or difference of sine or cosine:
`sin "A"/8 sin (3"A")/8`
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
Evaluate:
sin 50° – sin 70° + sin 10°
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 tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:
If secx cos5x + 1 = 0, where 0 < x ≤ `pi/2`, then find the value of x.
