Advertisements
Advertisements
प्रश्न
Prove that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
Advertisements
उत्तर
Consider LHS:
\[\sin (B - C) \cos (A - D) + \sin (C - A) \cos (B - D) + \sin (A - B) \cos (C - D)\]
Multiplying by 2:
\[ 2\sin (B - C) \cos (A - D) + 2\sin(C - A) \cos (B - D) + 2\sin (A - B) \cos (C - D)\]
\[ = \sin \left( B - C + A - D \right) + \sin \left( B - C - A + D \right) + \sin \left( C - A + B - D \right) + \sin \left( C - A - B + D \right) + \sin \left( A - B + C - D \right) + \sin \left( A - B - C + D \right)\]
\[ = \sin\left\{ - \left( C + D - A - B \right) \right\} + \sin\left\{ - \left( A + C - B - D \right) \right\} + \sin\left\{ - \left( A + D - B - C \right) \right\} + \sin\left( C - A - B + D \right) + \sin\left( A - B + C - D \right) + \sin\left( A - B - C + D \right)\]
\[ = - \sin\left( C + D - A - B \right) - \sin\left( A + C - B - D \right) - \sin\left( A + D - B - C \right) + \sin\left( C - A - B + D \right) + \sin\left( A - B + C - D \right) + \sin\left( A - B - C + D \right)\]
\[ = 0\]
= RHS
Hence, LHS = RHS.
APPEARS IN
संबंधित प्रश्न
Prove that:
Show that :
Prove that tan 20° tan 30° tan 40° tan 80° = 1.
Prove that:
sin 20° sin 40° sin 60° sin 80° = \[\frac{3}{16}\]
Show that:
sin A sin (B − C) + sin B sin (C − A) + sin C sin (A − B) = 0
Show that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
Prove that:
cos 100° + cos 20° = cos 40°
Prove that:
sin 105° + cos 105° = cos 45°
Prove that:
cos 55° + cos 65° + cos 175° = 0
Prove that:
cos 20° + cos 100° + cos 140° = 0
Prove that:
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:
If cosec A + sec A = cosec B + sec B, prove that tan A tan B = \[\cot\frac{A + B}{2}\].
If y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ.
Write the value of \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
The value of cos 52° + cos 68° + cos 172° is
If A, B, C are in A.P., then \[\frac{\sin A - \sin C}{\cos C - \cos A}\]=
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 =
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 7θ sin 3θ
Express the following as the product of sine and cosine.
cos 2A + cos 4A
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that:
(cos α – cos β)2 + (sin α – sin β)2 = 4 sin2 `((alpha - beta)/2)`
Prove that:
sin (A – B) sin C + sin (B – C) sin A + sin(C – A) sin B = 0
Prove that:
`(cos 7"A" +cos 5"A")/(sin 7"A" −sin 5"A")` = cot A
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.
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)]`
