Advertisements
Advertisements
प्रश्न
Show that:
sin A sin (B − C) + sin B sin (C − A) + sin C sin (A − B) = 0
Advertisements
उत्तर
Consider LHS:
\[\sin A \sin \left( B - C \right) + \sin B \sin \left( C - A \right) + \sin C \sin \left( A - B \right)\]
\[ = \frac{1}{2}\left[ 2\sin A \sin \left( B - C \right) \right] + \frac{1}{2}\left[ 2\sin B \sin \left( C - A \right) \right] + \frac{1}{2}\left[ 2\sin C \sin \left( A - B \right) \right]\]
\[ = \frac{1}{2}\left[ \cos \left\{ A - \left( B - C \right) \right\} - \cos \left\{ A + \left( B - C \right) \right\} \right] + \frac{1}{2}\left[ \cos \left\{ B - \left( C - A \right) \right\} - \cos \left\{ B + \left( C - A \right) \right\} \right] + \frac{1}{2}\left[ \cos \left\{ C - \left( A - B \right) \right\} - \cos \left\{ C + \left( A - B \right) \right\} \right]\]
\[ = \frac{1}{2}\left[ \cos \left( A - B + C \right) - \cos \left( A + B - C \right) \right] + \frac{1}{2}\left[ \cos \left( B - C + A \right) - \cos\left( B + C - A \right) \right] + \frac{1}{2}\left[ \cos\left( C - A + B \right) - \cos\left( C + A - B \right) \right]\]
\[ = \frac{1}{2}\cos\left( A - B + C \right) - \frac{1}{2}\cos \left( A + B - C \right) + \frac{1}{2}\cos \left( B - C + A \right) - \frac{1}{2}\cos \left( B + C - A \right) + \frac{1}{2}\cos \left( C - A + B \right) - \frac{1}{2}\cos\left( C + A - B \right)\]
\[ = \frac{1}{2}\cos\left( A - B + C \right) - \frac{1}{2}\cos\left( A + B - C \right) + \frac{1}{2}\cos\left( A + B - C \right) - \frac{1}{2}\cos\left( B + C - A \right) + \frac{1}{2}\cos\left( B + C - A \right) - \frac{1}{2}\cos\left( A - B + C \right)\]
\[ = 0\]
= RHS
APPEARS IN
संबंधित प्रश्न
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
If α + β = \[\frac{\pi}{2}\], show that the maximum value of cos α cos β is \[\frac{1}{2}\].
Express each of the following as the product of sines and cosines:
sin 5x − sin x
Prove that:
sin 38° + sin 22° = sin 82°
Prove that:
cos 100° + cos 20° = cos 40°
Prove that:
sin 23° + sin 37° = cos 7°
Prove that:
sin 40° + sin 20° = cos 10°
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
\[\sin\frac{5\pi}{18} - \cos\frac{4\pi}{9} = \sqrt{3} \sin\frac{\pi}{9}\]
Prove that:
Prove that:
cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A
Prove that:
`sin A + sin 2A + sin 4A + sin 5A = 4 cos (A/2) cos((3A)/2)sin3A`
Prove that:
sin 3A + sin 2A − sin A = 4 sin A cos \[\frac{A}{2}\] \[\frac{3A}{2}\]
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
If y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ.
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
Write the value of \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
Write the value of \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]
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 sin 2 θ + sin 2 ϕ = \[\frac{1}{2}\] and cos 2 θ + cos 2 ϕ = \[\frac{3}{2}\], then cos2 (θ − ϕ) =
The value of sin 50° − sin 70° + sin 10° is equal to
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)
Prove that:
cos 20° cos 40° cos 80° = `1/8`
Prove that:
sin (A – B) sin C + sin (B – C) sin A + sin(C – A) sin B = 0
Prove that cos 20° cos 40° cos 60° cos 80° = `3/16`.
If tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:
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)]`
