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:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
Show that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
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:
sin 47° + cos 77° = cos 17°
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ
Write the value of sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
If A + B = \[\frac{\pi}{3}\] and cos A + cos B = 1, then find the value of cos \[\frac{A - B}{2}\].
Write the value of \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
If sin 2 θ + sin 2 ϕ = \[\frac{1}{2}\] and cos 2 θ + cos 2 ϕ = \[\frac{3}{2}\], then cos2 (θ − ϕ) =
If sin α + sin β = a and cos α − cos β = b, then tan \[\frac{\alpha - \beta}{2}\]=
cos 35° + cos 85° + cos 155° =
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 \[\frac{\sin A - \sin C}{\cos C - \cos A}\]=
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 (7"A")/3 sin (5"A")/3`
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 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
Evaluate-
cos 20° + cos 100° + cos 140°
