Advertisements
Advertisements
प्रश्न
If cos A = m cos B, then write the value of \[\cot\frac{A + B}{2} \cot\frac{A - B}{2}\].
Advertisements
उत्तर
Given:
\[\cos A = m\cos B\]
\[ \Rightarrow \frac{\cos A}{\cos B} = \frac{m}{1}\]
\[ \Rightarrow \frac{\cos A + \cos B}{\cos A - \cos B} = \frac{m + 1}{m - 1}\]
\[ \Rightarrow \frac{2\cos\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right)}{- 2\sin\left( \frac{A + B}{2} \right)\sin\left( \frac{A - B}{2} \right)} = \frac{m + 1}{m - 1} \left[ \because \cos A + \cos B = 2\cos\left( \frac{A - B}{2} \right)\cos\left( \frac{A + B}{2} \right)\text{ and }\cos A - \cos B = - 2\sin\left( \frac{A + B}{2} \right)\cos\left( \frac{A - B}{2} \right) \right]\]
\[ \Rightarrow \frac{\cos\left( \frac{A - B}{2} \right)\cos\left( \frac{A + B}{2} \right)}{- \sin\left( \frac{A + B}{2} \right)\sin\left( \frac{A - B}{2} \right)} = \frac{m + 1}{m - 1} \]
\[ \Rightarrow -\cot\left( \frac{A + B}{2} \right)\cot\left( \frac{A - B}{2} \right)=\frac{m + 1}{m - 1}\]
APPEARS IN
संबंधित प्रश्न
Show that :
Prove that:
cos 40° cos 80° cos 160° = \[- \frac{1}{8}\]
Prove that:
sin 20° sin 40° sin 80° = \[\frac{\sqrt{3}}{8}\]
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
Prove that:
\[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x\]
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 5x − sin x
Express each of the following as the product of sines and cosines:
cos 12x - cos 4x
Prove that:
sin 23° + sin 37° = cos 7°
Prove that:
cos 55° + cos 65° + cos 175° = 0
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
cos 20° + cos 100° + cos 140° = 0
Prove that:
Prove that:
Prove that:
sin 47° + cos 77° = cos 17°
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:
cos (A + B + C) + cos (A − B + C) + cos (A + B − C) + cos (− A + B + C) = 4 cos A cos Bcos C
Prove that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
If A + B = \[\frac{\pi}{3}\] and cos A + cos B = 1, then find the value of cos \[\frac{A - B}{2}\].
The value of cos 52° + cos 68° + cos 172° is
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 =
Express the following as the sum or difference of sine or cosine:
`cos (7"A")/3 sin (5"A")/3`
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that:
tan 20° tan 40° tan 80° = `sqrt3`.
If cos A + cos B = `1/2` and sin A + sin B = `1/4`, prove that tan `(("A + B")/2) = 1/2`
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.
