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:
Show that :
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
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 tan 20° tan 30° tan 40° tan 80° = 1.
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:
sin 12x + sin 4x
Prove that:
sin 50° + sin 10° = cos 20°
Prove that:
sin 23° + sin 37° = cos 7°
Prove that:
sin 105° + cos 105° = cos 45°
Prove that:
sin 40° + sin 20° = cos 10°
Prove that:
cos 55° + cos 65° + cos 175° = 0
Prove that:
\[\sin\frac{5\pi}{18} - \cos\frac{4\pi}{9} = \sqrt{3} \sin\frac{\pi}{9}\]
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (− A + B + C) = 4 cos A cos Bcos C
If cosec A + sec A = cosec B + sec B, prove that tan A tan B = \[\cot\frac{A + B}{2}\].
Write the value of sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
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 \[\frac{A - B}{2}\].
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.
sin 163° cos 347° + sin 73° sin 167° =
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that:
cos 20° cos 40° cos 80° = `1/8`
Prove that:
`(cos 2"A" - cos 3"A")/(sin "2A" + sin "3A") = tan "A"/2`
Find the value of tan22°30′. `["Hint:" "Let" θ = 45°, "use" tan theta/2 = (sin theta/2)/(cos theta/2) = (2sin theta/2 cos theta/2)/(2cos^2 theta/2) = sintheta/(1 + costheta)]`
