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
Prove that:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]
Prove that:
sin 38° + sin 22° = sin 82°
Prove that:
sin 50° + sin 10° = cos 20°
Prove that:
sin 40° + sin 20° = cos 10°
Prove that:
cos 55° + cos 65° + cos 175° = 0
Prove that:
cos 80° + cos 40° − cos 20° = 0
Prove that:
cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A
Prove that:
cos 20° cos 100° + cos 100° cos 140° − 140° cos 200° = −\[\frac{3}{4}\]
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
If y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ.
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), prove that tan A tan B tan C + tan D = 0.
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 \[m \sin\theta = n \sin\left( \theta + 2\alpha \right)\], prove that \[\tan\left( \theta + \alpha \right) \cot\alpha = \frac{m + n}{m - n}\]
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.
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
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 \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]
cos 40° + cos 80° + cos 160° + cos 240° =
The value of cos 52° + cos 68° + cos 172° is
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:
`sin "A"/8 sin (3"A")/8`
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 2θ – cos θ
Prove that:
`(cos 2"A" - cos 3"A")/(sin "2A" + sin "3A") = tan "A"/2`
Prove that cos 20° cos 40° cos 60° cos 80° = `3/16`.
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 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:
