Advertisements
Advertisements
Question
Prove that:
Advertisements
Solution
Consider LHS:
\[ \frac{\sin\left( \theta + \phi \right) - 2\sin\theta + \sin\left( \theta - \phi \right)}{\cos\left( \theta + \phi \right) - 2\cos\theta + \cos\left( \theta - \phi \right)}\]
\[ = \frac{\sin\left( \theta + \phi \right) + \sin\left( \theta - \phi \right) - 2\sin\theta}{\cos\left( \theta + \phi \right) + \cos\left( \theta - \phi \right) - 2\cos\theta}\]
\[ = \frac{2\sin\left( \frac{\theta + \phi + \theta - \phi}{2} \right)\cos\left( \frac{\theta + \phi - \theta + \phi}{2} \right) - 2\sin\theta}{2\cos\left( \frac{\theta + \phi + \theta - \phi}{2} \right)\cos\left( \frac{\theta + \phi - \theta + \phi}{2} \right) - 2\cos\theta} \]
\[ = \frac{2\sin\theta\cos\phi - 2\sin\theta}{2\cos\theta\cos\phi - 2\cos\theta}\]
\[ = \frac{2\sin\theta\left[ \cos\phi - 1 \right]}{2\cos\theta\left[ \cos\phi - 1 \right]}\]
\[ = \tan\theta\]
= RHS
Hence, RHS = LHS.
APPEARS IN
RELATED QUESTIONS
Prove that:
Show that :
Prove that:
sin 20° sin 40° sin 80° = \[\frac{\sqrt{3}}{8}\]
Prove that:
sin 20° sin 40° sin 60° sin 80° = \[\frac{3}{16}\]
Show that:
sin A sin (B − C) + sin B sin (C − A) + sin C sin (A − B) = 0
Prove that:
\[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x\]
Express each of the following as the product of sines and cosines:
sin 12x + sin 4x
Express each of the following as the product of sines and cosines:
cos 12x - cos 4x
Prove that:
sin 38° + sin 22° = sin 82°
Prove that:
sin 23° + sin 37° = cos 7°
Prove that:
Prove that:
sin 47° + cos 77° = cos 17°
Prove that:
`sin A + sin 2A + sin 4A + sin 5A = 4 cos (A/2) cos((3A)/2)sin3A`
Prove that:
cos 20° cos 100° + cos 100° cos 140° − 140° cos 200° = −\[\frac{3}{4}\]
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 = \[\cot\frac{A + B}{2}\].
Prove that:
cos 40° + cos 80° + cos 160° + cos 240° =
The value of sin 50° − sin 70° + sin 10° is equal to
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:
cos(60° + A) sin(120° + A)
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 α – cos β)2 + (sin α – sin β)2 = 4 sin2 `((alpha - beta)/2)`
Prove that:
`(cos 2"A" - cos 3"A")/(sin "2A" + sin "3A") = tan "A"/2`
Prove that:
`(cos 7"A" +cos 5"A")/(sin 7"A" −sin 5"A")` = cot A
Prove that cos 20° cos 40° cos 60° cos 80° = `3/16`.
Evaluate:
sin 50° – sin 70° + sin 10°
If cosec A + sec A = cosec B + sec B prove that cot`(("A + B"))/2` = tan A tan B.
