Advertisements
Advertisements
प्रश्न
Prove that:
Advertisements
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:
Show that :
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
Prove that:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]
Prove that tan 20° tan 30° tan 40° tan 80° = 1.
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
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:
sin 12x + sin 4x
Express each of the following as the product of sines and cosines:
sin 5x − sin x
Express each of the following as the product of sines and cosines:
cos 12x + cos 8x
Prove that:
sin 40° + sin 20° = cos 10°
Prove that:
cos 55° + cos 65° + cos 175° = 0
Prove that:
Prove that:
Prove that:
cos 3A + cos 5A + cos 7A + cos 15A = 4 cos 4A cos 5A cos 6A
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
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 (cos α + cos β)2 + (sin α + sin β)2 = \[\lambda \cos^2 \left( \frac{\alpha - \beta}{2} \right)\], write the value of λ.
sin 163° cos 347° + sin 73° sin 167° =
If sin 2 θ + sin 2 ϕ = \[\frac{1}{2}\] and cos 2 θ + cos 2 ϕ = \[\frac{3}{2}\], then cos2 (θ − ϕ) =
cos 35° + cos 85° + cos 155° =
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 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
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
Prove that:
cos 20° cos 40° cos 80° = `1/8`
Prove that:
tan 20° tan 40° tan 80° = `sqrt3`.
