Advertisements
Advertisements
Question
Prove that:
cos 20° cos 40° cos 80° = `1/8`
Advertisements
Solution
cos 20° cos 40° cos 80°
= `((2 sin 20^circ)/(2 sin 20^circ))` cos 20° cos 40° cos 80°
[multiply and divide by 2 sin 20°]
`= ((2 sin 20^circ cos 20^circ) cos 40^circ cos 80^circ)/(2 sin 20^circ)`
`= (sin (2 xx 20^circ) cos 40^circ cos 80^circ)/(2 sin 20^circ)`
= `(sin 40^circ cos 40^circ cos 80^circ)/(2 sin 20^circ)`
(Multiply and divide by 2)
`= 1/2 xx ((2 sin 40^circ cos 40^circ))/(2 sin 20^circ) cos 80^circ`
`= 1/2 xx ((sin 2 xx 40^circ)cos 80^circ)/(2 sin 20^circ)`
`= 1/2 xx (sin 80^circ cos 80^circ)/(2 sin 20^circ)`
`= 1/2 xx 1/2 ((2 sin 80^circ cos 80^circ))/(2 sin 20^circ)`
`= 1/8 xx (sin 160^circ)/(sin 20^circ)`
`= 1/8 xx sin (180^circ - 20^circ)/(sin 20^circ)`
`= 1/8 xx sin 20^circ/sin 20^circ` ...[∵ sin(180° – θ) = sin θ]
`= 1/8 xx 1 = 1/8`
APPEARS IN
RELATED QUESTIONS
Prove that:
Prove that:
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
Prove that:
cos 20° cos 100° + cos 100° cos 140° − 140° cos 200° = −\[\frac{3}{4}\]
Prove that:
Prove that:
If \[x \cos\theta = y \cos\left( \theta + \frac{2\pi}{3} \right) = z \cos\left( \theta + \frac{4\pi}{3} \right)\], prove that \[xy + yz + zx = 0\]
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 sin A + sin B = α and cos A + cos B = β, then write the value of tan \[\left( \frac{A + B}{2} \right)\].
Express the following as the product of sine and cosine.
cos 2θ – cos θ
