Advertisements
Advertisements
प्रश्न
Prove that:
Advertisements
उत्तर
Consider LHS:
\[ \frac{\sin 11A \sin A + \sin 7A \sin 3A}{\cos 2A \sin A + \cos 6A \sin 3A}\]
Multiplying numerator and denominator by 2, we get
\[ = \frac{2\sin 11A \sin A + 2\sin 7A \sin 3A}{2\cos 11A sin A + 2\cos 7A \sin 3A}\]
\[ = \frac{\cos \left( 11A - A \right) - \cos \left( 11A + A \right) + \cos \left( 7A - 3A \right) - \cos \left( 7A + 3A \right)}{\sin \left( 11A + A \right) - \sin \left( 11A - A \right) + \sin \left( 7A + 3A \right) - \sin \left( 7A - 3A \right)}\]
\[ = \frac{\cos 10A - \cos 12A + \cos 4A - \cos 10A}{\sin 12A - \sin 10A + \sin 10A - \sin 4A}\]
\[ = \frac{\cos 4A - \cos 12A}{\sin 12A - \sin 4A}\]
\[ = \frac{- 2\sin \left( \frac{4A + 12A}{2} \right) \sin \left( \frac{4A - 12A}{2} \right)}{2\sin \left( \frac{12A - 4A}{2} \right) \cos \left( \frac{12A + 4A}{2} \right)}\]
\[ = \frac{- \sin 8A \sin \left( - 4A \right)}{\sin 4A \cos 8A}\]
\[ = \frac{\sin 8A \sin 4A}{\sin 4A \cos 8A}\]
\[ = \tan8A\]
= RHS
Hence, LHS = RHS.
APPEARS IN
संबंधित प्रश्न
Show that :
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
Prove that:
tan 20° tan 40° tan 60° tan 80° = 3
If α + β = \[\frac{\pi}{2}\], show that the maximum value of cos α cos β is \[\frac{1}{2}\].
Express each of the following as the product of sines and cosines:
sin 12x + sin 4x
Prove that:
sin 38° + sin 22° = sin 82°
Prove that:
cos 100° + cos 20° = cos 40°
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:
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:
`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:
If y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ.
If \[x \cos\theta = y \cos\left( \theta + \frac{2\pi}{3} \right) = z \cos\left( \theta + \frac{4\pi}{3} \right)\], prove that \[xy + yz + zx = 0\]
Write the value of sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
sin 163° cos 347° + sin 73° sin 167° =
If sin 2 θ + sin 2 ϕ = \[\frac{1}{2}\] and cos 2 θ + cos 2 ϕ = \[\frac{3}{2}\], then cos2 (θ − ϕ) =
cos 35° + cos 85° + cos 155° =
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 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`
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
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.
