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
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 tan 20° tan 30° tan 40° tan 80° = 1.
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
Show that:
sin A sin (B − C) + sin B sin (C − A) + sin C sin (A − B) = 0
Show that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
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 5x − sin x
Express each of the following as the product of sines and cosines:
sin 2x + cos 4x
Prove that:
sin 23° + sin 37° = cos 7°
Prove that:
cos 20° + cos 100° + cos 140° = 0
Prove that:
\[\sin\frac{5\pi}{18} - \cos\frac{4\pi}{9} = \sqrt{3} \sin\frac{\pi}{9}\]
Prove that:
Prove that:
sin 47° + cos 77° = cos 17°
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 (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 \[\left( \frac{A + B}{2} \right)\].
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
If sin 2A = λ sin 2B, then write the value of \[\frac{\lambda + 1}{\lambda - 1}\]
Write the value of \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]
The value of cos 52° + cos 68° + cos 172° is
sin 47° + sin 61° − sin 11° − sin 25° is equal to
If \[\tan\alpha = \frac{x}{x + 1}\] and
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that:
(cos α – cos β)2 + (sin α – sin β)2 = 4 sin2 `((alpha - beta)/2)`
Prove that:
2 cos `pi/13` cos \[\frac{9\pi}{13} + \text{cos} \frac{3\pi}{13} + \text{cos} \frac{5\pi}{13}\] = 0
Evaluate-
cos 20° + cos 100° + cos 140°
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.
If tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:
