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.
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:
Prove that:
tan 20° tan 40° tan 60° tan 80° = 3
Prove that
\[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x\]
Express each of the following as the product of sines and cosines:
sin 2x + cos 4x
Prove that:
sin 50° + sin 10° = cos 20°
Prove that:
sin 105° + cos 105° = cos 45°
Prove that:
cos 80° + cos 40° − cos 20° = 0
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:
Prove that:
Prove that:
If cosec A + sec A = cosec B + sec B, prove that tan A tan B = \[\cot\frac{A + B}{2}\].
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 \[m \sin\theta = n \sin\left( \theta + 2\alpha \right)\], prove that \[\tan\left( \theta + \alpha \right) \cot\alpha = \frac{m + n}{m - n}\]
If sin A + sin B = α and cos A + cos B = β, then write the value of tan \[\left( \frac{A + B}{2} \right)\].
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
If sin 2A = λ sin 2B, then write the value of \[\frac{\lambda + 1}{\lambda - 1}\]
If cos (A + B) sin (C − D) = cos (A − B) sin (C + D), then write the value of tan A tan B tan C.
sin 163° cos 347° + sin 73° sin 167° =
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
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 7θ sin 3θ
Express the following as the product of sine and cosine.
sin 6θ – sin 2θ
Prove that:
cos 20° cos 40° cos 80° = `1/8`
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 tan θ = `1/sqrt5` and θ lies in the first quadrant then cos θ is:
Find the value of tan22°30′. `["Hint:" "Let" θ = 45°, "use" tan theta/2 = (sin theta/2)/(cos theta/2) = (2sin theta/2 cos theta/2)/(2cos^2 theta/2) = sintheta/(1 + costheta)]`
