Advertisements
Advertisements
Question
Show that:
sin A sin (B − C) + sin B sin (C − A) + sin C sin (A − B) = 0
Advertisements
Solution
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
RELATED QUESTIONS
Prove that:
cos 40° cos 80° cos 160° = \[- \frac{1}{8}\]
Prove that:
tan 20° tan 40° tan 60° tan 80° = 3
Prove that:
sin 20° sin 40° sin 60° sin 80° = \[\frac{3}{16}\]
Express each of the following as the product of sines and cosines:
cos 12x - cos 4x
Prove that:
sin 38° + sin 22° = sin 82°
Prove that:
sin 23° + sin 37° = cos 7°
Prove that:
sin 105° + cos 105° = cos 45°
Prove that:
cos 80° + cos 40° − cos 20° = 0
Prove that:
\[\sin\frac{5\pi}{18} - \cos\frac{4\pi}{9} = \sqrt{3} \sin\frac{\pi}{9}\]
Prove that:
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:
Prove that:
cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (− A + B + C) = 4 cos A cos Bcos C
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 \[m \sin\theta = n \sin\left( \theta + 2\alpha \right)\], prove that \[\tan\left( \theta + \alpha \right) \cot\alpha = \frac{m + n}{m - n}\]
If 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 \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.
If sin α + sin β = a and cos α − cos β = b, then tan \[\frac{\alpha - \beta}{2}\]=
sin 47° + sin 61° − sin 11° − sin 25° is equal to
If cos A = m cos B, then \[\cot\frac{A + B}{2} \cot\frac{B - A}{2}\]=
If sin x + sin y = \[\sqrt{3}\] (cos y − cos x), then sin 3x + sin 3y =
If \[\tan\alpha = \frac{x}{x + 1}\] and
Express the following as the sum or difference of sine or cosine:
cos(60° + A) sin(120° + A)
Express the following as the product of sine and cosine.
sin 6θ – sin 2θ
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:
