Advertisements
Advertisements
Question
Show that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
Advertisements
Solution
Consider LHS:
\[\sin \left( B - C \right) \cos \left( A - D \right) + \sin \left( C - A \right) \cos \left( B - D \right) + \sin \left( A - B \right) \cos \left( C - D \right)\]
\[= \frac{1}{2}\left[ 2\sin \left( B - C \right) \cos \left( A - D \right) \right] + \frac{1}{2}\left[ 2\sin \left( C - A \right) \cos \left( B - D \right) \right] + \frac{1}{2}\left[ 2\sin \left( A - B \right) \cos\left( C - D \right) \right]\]
\[ = \frac{1}{2}\left[ \sin \left\{ \left( B - C \right) + \left( A - D \right) \right\} + \sin \left\{ \left( B - C \right) - \left( A - D \right) \right\} \right] + \frac{1}{2}\left[ \sin \left\{ \left( C - A \right) + \left( B - D \right) \right\} + \sin \left\{ \left( C - A \right) - \left( B - D \right) \right\} \right] + \frac{1}{2}\left[ \sin \left\{ \left( A - B \right) + \left( C - D \right) \right\} + \sin \left\{ \left( A - B \right) - \left( C - D \right) \right\} \right]\]
\[ = \frac{1}{2}\left[ \sin \left( B - C + A - D \right) + \sin \left( B - C - A + D \right) \right] + \frac{1}{2}\left[ \sin \left( C - A + B - D \right) + \sin \left( C - A - B + D \right) \right] + \frac{1}{2}\left[ \sin \left( A - B + C - D \right) + \sin \left( A - B - C + D \right) \right]\]
\[ = \frac{1}{2}\left[ \sin \left( B - C + A - D \right) + \sin \left( B - C - A + D \right) \right] + \frac{1}{2}\left[ \sin \left\{ - \left( - C + A - B + D \right) \right\} + \sin \left\{ - \left( - C + A + B - D \right) \right\} \right] + \frac{1}{2}\left[ \sin\left\{ - \left( - A + B - C + D \right) \right\} + \sin \left( A - B - C + D \right) \right]\]
\[ = \frac{1}{2}\sin\left( B - C + A - D \right) + \frac{1}{2}\sin\left( B - C - A + D \right) - \frac{1}{2}\sin\left( - C + A - B + D \right) - \frac{1}{2}\sin\left( - C + A + B - D \right) - \frac{1}{2}\sin\left( - A + B - C + D \right) + \frac{1}{2}\sin\left( A - B - C + D \right)\]
\[ = 0\]
= RHS
APPEARS IN
RELATED QUESTIONS
Prove that:
Show that :
Prove that:
sin 20° sin 40° sin 80° = \[\frac{\sqrt{3}}{8}\]
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 2x + cos 4x
Prove that:
sin 38° + sin 22° = sin 82°
Prove that:
cos 55° + cos 65° + cos 175° = 0
Prove that:
Prove that:
Prove that:
sin 47° + cos 77° = cos 17°
Prove that:
`sin A + sin 2A + sin 4A + sin 5A = 4 cos (A/2) cos((3A)/2)sin3A`
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 \[x \cos\theta = y \cos\left( \theta + \frac{2\pi}{3} \right) = z \cos\left( \theta + \frac{4\pi}{3} \right)\], prove that \[xy + yz + zx = 0\]
If \[m \sin\theta = n \sin\left( \theta + 2\alpha \right)\], prove that \[\tan\left( \theta + \alpha \right) \cot\alpha = \frac{m + n}{m - n}\]
Write the value of sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
If cos A = m cos B, then write the value of \[\cot\frac{A + B}{2} \cot\frac{A - B}{2}\].
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.
The value of cos 52° + cos 68° + cos 172° is
If sin α + sin β = a and cos α − cos β = b, then tan \[\frac{\alpha - \beta}{2}\]=
The value of sin 50° − sin 70° + sin 10° is equal to
If cos A = m cos B, then \[\cot\frac{A + B}{2} \cot\frac{B - A}{2}\]=
If sin (B + C − A), sin (C + A − B), sin (A + B − C) are in A.P., then cot A, cot B and cot Care in
If sin x + sin y = \[\sqrt{3}\] (cos y − cos x), then sin 3x + sin 3y =
Prove that:
sin A sin(60° + A) sin(60° – A) = `1/4` sin 3A
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 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)]`
If secx cos5x + 1 = 0, where 0 < x ≤ `pi/2`, then find the value of x.
