Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[\text{ LHS }= \sin\frac{x}{2}\sin\frac{7x}{2} + \sin\frac{3x}{2}\sin\frac{11x}{2}\]
\[ = \frac{1}{2}\left[ 2\sin\frac{x}{2}\sin\frac{7x}{2} + 2\sin\frac{3x}{2}\sin\frac{11x}{2} \right]\]
\[ = \frac{1}{2}\left[ \cos\left( \frac{7x}{2} - \frac{x}{2} \right) - \cos\left( \frac{7x}{2} + \frac{x}{2} \right) + \cos\left( \frac{11x}{2} - \frac{3x}{2} \right) - \cos\left( \frac{11x}{2} + \frac{3x}{2} \right) \right]\]
\[ = \frac{1}{2}\left[ \cos3x - \cos4x + \cos4x - \cos7x \right]\]
\[ = \frac{1}{2}\left[ \cos3x - \cos7x \right]\]
\[ = \frac{1}{2}\left[ - 2\sin\left( \frac{3x + 7x}{2} \right)\sin\left( \frac{3x - 7x}{2} \right) \right]\]
\[ = \frac{1}{2}\left[ - 2\sin\left( 5x \right)\sin\left( - 2x \right) \right]\]
\[ = \sin\left( 5x \right)\sin\left( 2x \right) =\text{ RHS }\]
Hence, LHS = RHS
APPEARS IN
संबंधित प्रश्न
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
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
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
Prove that:
Prove that:
sin 47° + cos 77° = cos 17°
Prove that:
cos A + cos 3A + cos 5A + cos 7A = 4 cos A cos 2A cos 4A
Prove that:
Prove that:
Prove that:
Prove that:
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
Prove that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
If cos (α + β) sin (γ + δ) = cos (α − β) sin (γ − δ), prove that cot α cot β cot γ = cot δ
If \[x \cos\theta = y \cos\left( \theta + \frac{2\pi}{3} \right) = z \cos\left( \theta + \frac{4\pi}{3} \right)\], prove that \[xy + yz + zx = 0\]
Write the value of sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
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 \[\frac{A - B}{2}\].
If sin 2A = λ sin 2B, then write the value of \[\frac{\lambda + 1}{\lambda - 1}\]
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.
cos 40° + cos 80° + cos 160° + cos 240° =
If cos A = m cos B, then \[\cot\frac{A + B}{2} \cot\frac{B - A}{2}\]=
If \[\tan\alpha = \frac{x}{x + 1}\] and
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.
sin A + sin 2A
Express the following as the product of sine and cosine.
cos 2A + cos 4A
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.
Find the value of tan22°30′. `["Hint:" "Let" θ = 45°, "use" tan theta/2 = (sin theta/2)/(cos theta/2) = (2sin theta/2 cos theta/2)/(2cos^2 theta/2) = sintheta/(1 + costheta)]`
If secx cos5x + 1 = 0, where 0 < x ≤ `pi/2`, then find the value of x.
