Advertisements
Advertisements
प्रश्न
Prove that:
उत्तर
Consider LHS:
\[ \frac{\cos(A + B + C) + \cos( - A + B + C) + \cos(A - B + C) + \cos(A + B - C)}{\sin(A + B + C) + \sin( - A + B + C) + \sin(A - B + C) - \sin(A + B - C)}\]
\[ = \frac{2\cos\left( \frac{A + B + C - A + B + C}{2} \right)\cos\left( \frac{A + B + C + A - B - C}{2} \right) + 2\cos\left( \frac{A - B + C + A + B - C}{2} \right)\cos\left( \frac{A - B + C - A - B + C}{2} \right)}{2\sin\left( \frac{A + B + C - A + B + C}{2} \right)\cos\left( \frac{A + B + C + A - B - C}{2} \right) + 2\sin\left( \frac{A - B + C - A - B + C}{2} \right)\cos\left( \frac{A - B + C + A + B - C}{2} \right)}\]
\[ = \frac{2\cos \left( B + C \right) \cos A + 2\cos A \cos \left( - B + C \right)}{2\sin \left( B + C \right) \cos A + 2\sin \left( - B + C \right) \cos A}\]
\[ = \frac{2\cos A\left[ \cos \left( B + C \right) + \cos\left( - B + C \right) \right]}{2\cos A\left[ \sin\left( B + C \right) + \sin\left( - B + C \right) \right]}\]
\[ = \frac{\cos \left( B + C \right) + \cos \left( - B + C \right)}{\sin\left( B + C \right) + \sin \left( - B + C \right)}\]
\[ = \frac{2\cos \left( \frac{B + C - B + C}{2} \right) \cos \left( \frac{B + C + B - C}{2} \right)}{2\sin\left( \frac{B + C - B + C}{2} \right) \cos \left( \frac{B + C + B - C}{2} \right)}\]
\[ = \frac{\cos C \cos B}{\sin C \cos B}\]
\[ = \cot C\]
= RHS
Hence, LHS = RHS.
APPEARS IN
संबंधित प्रश्न
Prove that:
Prove that:
cos 40° cos 80° cos 160° = \[- \frac{1}{8}\]
Prove that:
tan 20° tan 40° tan 60° tan 80° = 3
Prove that tan 20° tan 30° tan 40° tan 80° = 1.
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
Express each of the following as the product of sines and cosines:
sin 12x + sin 4x
Express each of the following as the product of sines and cosines:
cos 12x + cos 8x
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:
sin 3A + sin 2A − sin A = 4 sin A cos \[\frac{A}{2}\] \[\frac{3A}{2}\]
Prove that:
cos 20° cos 100° + cos 100° cos 140° − 140° cos 200° = −\[\frac{3}{4}\]
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
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
Write the value of \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
cos 40° + cos 80° + cos 160° + cos 240° =
The value of sin 78° − sin 66° − sin 42° + sin 60° is ______.
If sin α + sin β = a and cos α − cos β = b, then tan \[\frac{\alpha - \beta}{2}\]=
cos 35° + cos 85° + cos 155° =
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 A, B, C are in A.P., then \[\frac{\sin A - \sin C}{\cos C - \cos A}\]=
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
Prove that:
sin (A – B) sin C + sin (B – C) sin A + sin(C – A) sin B = 0
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.
