Advertisements
Advertisements
Question
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.
Advertisements
Solution
cos (A + B) sin (C − D) = cos (A − B) sin (C + D)
\[\text{ Dividing both sides by }\cos A \cos B \cos C \cos D:\]
\[\frac{\left[ \cos A \cos B - \sin A \sin B \right]\left[ \sin C \cos D - \cos C \sin D \right]}{\cos A \cos B \cos C \cos D} = \frac{\left[ \cos A \cos B + \sin A \sin B \right]\left[ \sin C \cos D + \cos C \sin D \right]}{\cos A \cos B \cos C \cos D}\]
\[ \Rightarrow \frac{\left[ \cos A \cos B - \sin A \sin B \right]}{\cos A \cos B} \times \frac{\left[ \sin C \cos D - \cos C \sin D \right]}{\cos C \cos D} = \frac{\left[ \cos A \cos B + \sin A \sin B \right]}{\cos A \cos B} \times \frac{\left[ \sin C cos D + \cos C \sin D \right]}{\cos C \cos D}\]
\[ \Rightarrow \left[ 1 - \tan A \tan B \right]\left[ \tan C - \tan D \right] = \left[ 1 + \tan A \tan B \right]\left[ \tan C + \tan D \right]\]
\[ \Rightarrow \tan C - \tan D - \tan A \tan B \tan C + \tan A \tan B \tan D = \tan C + \tan D + \tan A \tan B \tan C + \tan A \tan B \tan D\]
\[ \Rightarrow - \tan D - \tan D = \tan A \tan B \tan C + \tan A \tan B \tan C\]
\[ \Rightarrow - 2\tan D = 2\tan A\tan B\tan C\]
\[ \Rightarrow \tan A\tan B\tan C = - \tan D\]
APPEARS IN
RELATED QUESTIONS
Show that :
Show that :
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:
\[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x\]
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:
sin 2x + cos 4x
Prove that:
sin 105° + cos 105° = cos 45°
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
cos 20° + cos 100° + cos 140° = 0
Prove that:
\[\sin\frac{5\pi}{18} - \cos\frac{4\pi}{9} = \sqrt{3} \sin\frac{\pi}{9}\]
Prove that:
Prove that:
Prove that:
`sin A + sin 2A + sin 4A + sin 5A = 4 cos (A/2) cos((3A)/2)sin3A`
Prove that:
If y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ.
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), prove that tan A tan B tan C + tan D = 0.
Write the value of sin \[\frac{\pi}{12}\] sin \[\frac{5\pi}{12}\].
If A + B = \[\frac{\pi}{3}\] and cos A + cos B = 1, then find the value of cos \[\frac{A - B}{2}\].
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° =
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) =
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 \[\frac{\alpha - \beta}{2}\]=
The value of sin 50° − sin 70° + sin 10° is equal to
sin 47° + sin 61° − sin 11° − sin 25° is equal to
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:
cos 7θ sin 3θ
Express the following as the product of sine and cosine.
sin A + sin 2A
Express the following as the product of sine and cosine.
sin 6θ – sin 2θ
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that:
cos 20° cos 40° cos 80° = `1/8`
If cos A + cos B = `1/2` and sin A + sin B = `1/4`, prove that tan `(("A + B")/2) = 1/2`
If secx cos5x + 1 = 0, where 0 < x ≤ `pi/2`, then find the value of x.
