Advertisements
Advertisements
Question
Prove that:
Advertisements
Solution
Consider LHS:
\[ \frac{\sin 3A \cos 4A - \sin A \cos 2A}{\sin 4A sin A + \cos 6A \cos A}\]
Multiplying numerator and denominator by 2, we get
\[ = \frac{2\sin 3A \cos 4A - 2\sin A \cos 2A}{2\sin 4A \sin A + 2\cos 6A \cos A}\]
\[ = \frac{\sin \left( 3A + 4A \right) + \sin \left( 3A - 4A \right) - \sin \left( A + 2A \right) - \sin \left( A - 2A \right)}{\cos \left( 4A - A \right) - \cos \left( 4A + A \right) + \cos \left( 6A + A \right) + \cos \left( 6A - A \right)}\]
\[ = \frac{\sin 7A + \sin \left( - A \right) - \sin 3A - \sin \left( - A \right)}{\cos 3A - \cos 5A + \cos 7A + \cos 5A}\]
\[ = \frac{\sin 7A - \sin A - \sin 3A + \sin A}{\cos 3A - \cos 5A + \cos 7A + \cos 5A}\]
\[ = \frac{\sin 7A - \sin 3A}{\cos 3A + \cos 7A}\]
\[ = \frac{2\sin \left( \frac{7A - 3A}{2} \right) \cos \left( \frac{7A + 3A}{2} \right)}{2\cos \left( \frac{3A + 7A}{2} \right) \cos \left( \frac{3A - 7A}{2} \right)}\]
\[ = \frac{\sin 2A \cos 5A}{\cos 5A \cos \left( - 2A \right)}\]
\[ = \frac{\sin 2A \cos 5A}{\cos 5A \cos 2A}\]
\[ = \tan 2A\]
= RHS
Hence, LHS = RHS.
APPEARS IN
RELATED QUESTIONS
Prove that:
Show 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:
sin 20° sin 40° sin 60° sin 80° = \[\frac{3}{16}\]
Prove that:
\[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x\]
Prove that:
sin 38° + sin 22° = sin 82°
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 y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ.
If \[m \sin\theta = n \sin\left( \theta + 2\alpha \right)\], prove that \[\tan\left( \theta + \alpha \right) \cot\alpha = \frac{m + n}{m - n}\]
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 \[\cot\frac{A + B}{2} \cot\frac{A - B}{2}\].
Write the value of \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.
cos 40° + cos 80° + cos 160° + cos 240° =
The value of cos 52° + cos 68° + cos 172° is
The value of sin 78° − sin 66° − sin 42° + sin 60° is ______.
sin 47° + sin 61° − sin 11° − sin 25° is equal to
If cos A = m cos B, then \[\cot\frac{A + B}{2} \cot\frac{B - A}{2}\]=
If A, B, C are in A.P., then \[\frac{\sin A - \sin C}{\cos C - \cos A}\]=
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 (7"A")/3 sin (5"A")/3`
Express the following as the product of sine and cosine.
sin A + sin 2A
Express the following as the product of sine and cosine.
sin 6θ – sin 2θ
Prove that:
sin A sin(60° + A) sin(60° – A) = `1/4` sin 3A
Prove that:
sin (A – B) sin C + sin (B – C) sin A + sin(C – A) sin B = 0
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 tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:
