Advertisements
Advertisements
प्रश्न
Prove that:
Advertisements
उत्तर
Consider LHS:
\[ \sin \alpha + \sin \beta + \sin \gamma - \sin (\alpha + \beta + \gamma)\]
\[ = 2sin\left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right) + 2\cos \left( \frac{\gamma + \alpha + \beta + \gamma}{2} \right) \sin \left( \frac{\gamma - \alpha - \beta - \gamma}{2} \right)\]
\[\]
\[ = 2\sin\left( \frac{\alpha + \beta}{2} \right)\cos\left( \frac{\alpha - \beta}{2} \right) + 2\cos\left( \frac{2\gamma + \alpha + \beta}{2} \right)\sin\left( \frac{- \alpha - \beta}{2} \right)\]
\[\]
\[ = 2\sin\left( \frac{\alpha + \beta}{2} \right)\cos\left( \frac{\alpha - \beta}{2} \right) + 2\cos\left( \frac{2\gamma + \alpha + \beta}{2} \right)\sin\left[ - \left( \frac{\alpha + \beta}{2} \right) \right]\]
\[\]
\[ = 2\sin\left( \frac{\alpha + \beta}{2} \right)\left[ \cos\left( \frac{\alpha - \beta}{2} \right) - \cos\left( \frac{2\gamma + \alpha + \beta}{2} \right) \right]\]
\[\]
\[ = 2\sin\left( \frac{\alpha + \beta}{2} \right)\left[ - 2\sin\left( \frac{\alpha - \beta + 2\gamma + \alpha + \beta}{4} \right) \sin\left( \frac{\alpha - \beta - 2\gamma - \alpha - \beta}{4} \right) \right]\]
\[\]
\[ = 2\sin\left( \frac{\alpha + \beta}{2} \right)\left[ - 2\sin\left( \frac{\alpha + \gamma}{2} \right) \sin\left( \frac{- \beta - \gamma}{2} \right) \right]\]
\[\]
\[ = 2\sin\left( \frac{\alpha + \beta}{2} \right)\left[ 2\sin\left( \frac{\alpha + \gamma}{2} \right) sin\left( \frac{\beta + \gamma}{2} \right) \right]\]
\[\]
\[ = 4\sin\left( \frac{\alpha + \beta}{2} \right) \sin\left( \frac{\alpha + \gamma}{2} \right) \sin\left( \frac{\beta + \gamma}{2} \right)\]
\[\]
= RHS
Hence, LHS = RHS.
APPEARS IN
संबंधित प्रश्न
Prove that:
Prove 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:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
If α + β = \[\frac{\pi}{2}\], show that the maximum value of cos α cos β is \[\frac{1}{2}\].
Prove that:
sin 38° + sin 22° = sin 82°
Prove that:
cos 100° + cos 20° = cos 40°
Prove that:
sin 105° + cos 105° = cos 45°
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 3A + sin 2A − sin A = 4 sin A cos \[\frac{A}{2}\] \[\frac{3A}{2}\]
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:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
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 \[xy + yz + zx = 0\]
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.
The value of sin 50° − sin 70° + sin 10° 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 x + sin y = \[\sqrt{3}\] (cos y − cos x), then sin 3x + sin 3y =
Express the following as the sum or difference of sine or cosine:
cos(60° + A) sin(120° + A)
Express the following as the sum or difference of sine or cosine:
cos 7θ sin 3θ
Express the following as the product of sine and cosine.
cos 2A + cos 4A
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that:
sin A sin(60° + A) sin(60° – A) = `1/4` sin 3A
Prove that cos 20° cos 40° cos 60° cos 80° = `3/16`.
Evaluate:
sin 50° – sin 70° + sin 10°
If cos A + cos B = `1/2` and sin A + sin B = `1/4`, prove that tan `(("A + B")/2) = 1/2`
If cosec A + sec A = cosec B + sec B prove that cot`(("A + B"))/2` = tan A tan B.
If tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:
