Advertisements
Advertisements
प्रश्न
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), prove that tan A tan B tan C + tan D = 0.
Advertisements
उत्तर
cos (A + B) sin (C − D) = cos (A − B) sin (C + D)
\[\Rightarrow\][cosA cosB − sinA sinB] [sinC cosD − cosC sinD] = [cosA cosB + sinA sinB] [sinC cosD + cosC sinD]
\[\text{ Dividing both sides by }\cos A \cos B \cos C \cos D, \]
\[\frac{\left[ \cos A \cos B - \sin A \sin B \right]\left[ \sin C \cos D - \cos C \sin D \right]}{\cos A \cos B \cos C \cos D} = \frac{\left[ \cos A \cos B + \sin A \sin B \right]\left[ \sin C \cos D + \cos C \sin D \right]}{\cos A \cos B \cos C \cos D}\]
\[ \Rightarrow \frac{\left[ \cos A \cos B - \sin A \sin B \right]}{\cos A \cos B} \times \frac{\left[ \sin C\cos D - \cos C \sin D \right]}{\cos C \cos D} = \frac{\left[ \cos A \cos B + \sin A \sin B \right]}{\cos A \cos B} \times \frac{\left[ \sin C \cos D + \cos C \sin D \right]}{\cos C \cos D}\]
\[ \Rightarrow \left[ 1 - \tan A \tan B \right]\left[ \tan C - \tan D \right] = \left[ 1 + \tan A \tan B \right]\left[ \tan C + \tan D \right]\]
\[ \Rightarrow \tan C - \tan D - \tan A \tan B \tan C + \tan A \tan B \tan D = \tan C + \tan D + \tan A \tan B \tan C + \tan A \tan B \tan D\]
\[ \Rightarrow - \tan D - \tan D = \tan A \tan B \tan C + \tan A \tan B \tan C\]
\[ \Rightarrow - 2\tan D = 2\tan A \tan B \tan C\]
\[ \Rightarrow \tan A \tan B \tan C = - \tan D\]
\[ \Rightarrow \tan A \tan B \tan C + \tan D = 0\]
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove that:
Show that :
Prove that:
sin 20° sin 40° sin 80° = \[\frac{\sqrt{3}}{8}\]
Prove that:
tan 20° tan 40° tan 60° tan 80° = 3
Prove that:
sin 20° sin 40° sin 60° sin 80° = \[\frac{3}{16}\]
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 12x + sin 4x
Prove that:
sin 38° + sin 22° = sin 82°
Prove that:
sin 50° + sin 10° = cos 20°
Prove that:
sin 23° + sin 37° = cos 7°
Prove that:
cos 55° + cos 65° + cos 175° = 0
Prove that:
cos 80° + cos 40° − cos 20° = 0
Prove that:
Prove that:
Prove that:
sin 47° + cos 77° = cos 17°
Prove that:
cos A + cos 3A + cos 5A + cos 7A = 4 cos A cos 2A cos 4A
Prove that:
`sin A + sin 2A + sin 4A + sin 5A = 4 cos (A/2) cos((3A)/2)sin3A`
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:
If \[x \cos\theta = y \cos\left( \theta + \frac{2\pi}{3} \right) = z \cos\left( \theta + \frac{4\pi}{3} \right)\], prove that \[xy + yz + zx = 0\]
If cos A = m cos B, then write the value of \[\cot\frac{A + B}{2} \cot\frac{A - B}{2}\].
Write the value of \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
If sin 2A = λ sin 2B, then write the value of \[\frac{\lambda + 1}{\lambda - 1}\]
Express the following as the sum or difference of sine or cosine:
cos(60° + A) sin(120° + A)
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.
sin A + sin 2A
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 secx cos5x + 1 = 0, where 0 < x ≤ `pi/2`, then find the value of x.
