Advertisements
Advertisements
प्रश्न
Prove that:
sin 3A + sin 2A − sin A = 4 sin A cos \[\frac{A}{2}\] \[\frac{3A}{2}\]
Advertisements
उत्तर
Consider LHS:
\[ = \sin 3A + \sin 2A - \sin A\]
\[ = 2\sin \left( \frac{3A + 2A}{2} \right) cos \left( \frac{3A - 2A}{2} \right) - \sin A \left\{ \because \sin A + \sin B = 2\sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \right\}\]
\[ = 2\sin \left( \frac{5}{2}A \right) \cos \left( \frac{A}{2} \right) - \sin A\]
\[= 2\sin \left( \frac{5}{2}A \right) cos \left( \frac{A}{2} \right) - 2\sin \frac{A}{2} \cos \frac{A}{2}\]
\[ = 2\cos \left( \frac{A}{2} \right) \left\{ \sin \frac{5}{2}A - \sin \frac{A}{2} \right\}\]
\[ = 2\cos \left( \frac{A}{2} \right) \times 2\sin \left( \frac{\frac{5}{2}A - \frac{A}{2}}{2} \right) \cos \left( \frac{\frac{5}{2}A + \frac{A}{2}}{2} \right)\]
\[ = 4\cos \left( \frac{A}{2} \right) \sin A \cos \left( \frac{3}{2}A \right)\]
\[ = 4\sin A \cos$\left( \frac{A}{2} \right)$\cos \left( \frac{3}{2}A \right)\]
= RHS
Hence, LHS = RHS
APPEARS IN
संबंधित प्रश्न
Show that :
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:
\[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x\]
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:
cos 12x + cos 8x
Prove that:
\[\sin\frac{5\pi}{18} - \cos\frac{4\pi}{9} = \sqrt{3} \sin\frac{\pi}{9}\]
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
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.
If sin A + sin B = α and cos A + cos B = β, then write the value of tan \[\left( \frac{A + B}{2} \right)\].
If cos A = m cos B, then write the value of \[\cot\frac{A + B}{2} \cot\frac{A - B}{2}\].
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 \[\frac{\alpha - \beta}{2}\]=
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:
`sin "A"/8 sin (3"A")/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°
If tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:
