Advertisements
Advertisements
प्रश्न
If A + B = C, then write the value of tan A tan B tan C.
Advertisements
उत्तर
\[\tan A \tan B \tan C = \tan A \tan B \tan(A + B) \left[ \text{ Using } A + B = C \right]\]
\[ = \tan A \tan B \times \frac{\tan A + \tan B}{1 - \tan A \tan B}\]
\[ = \frac{\tan^2 A\tan B + \tan A \tan^2 B}{1 - \tan A \tan B}\]
\[ = \frac{\tan^2 A\tan B + \tan A \tan^2 B + \tan A + \tan B - \tan A - \tan B}{1 - \tan A \tan B}\]
\[ = \frac{- \tan A(1 - \tan A\tan B) - \tan B(1 - \tan A\tan B) + \tan A + \tan B}{1 - \tan A \tan B}\]
\[ = \frac{- (1 - \tan A\tan B)\left( \tan A + \tan B \right) + \tan A + \tan B}{1 - \tan A \tan B}\]
\[ = \frac{\tan A + \tan B}{1 - \tan A \tan B} - \tan A - \tan B \]
\[ = \tan(A + B) - \tan A - \tan B\]
\[ = \tan C - \tan A - \tan B\]
\[\]
APPEARS IN
संबंधित प्रश्न
Prove that: `sin^2 pi/6 + cos^2 pi/3 - tan^2 pi/4 = -1/2`
Find the value of: tan 15°
Prove the following: `cos (pi/4 xx x) cos (pi/4 - y) - sin (pi/4 - x)sin (pi/4 - y) = sin (x + y)`
Prove the following:
cos2 2x – cos2 6x = sin 4x sin 8x
Prove the following:
`(cos9x - cos5x)/(sin17x - sin 3x) = - (sin2x)/(cos 10x)`
Prove the following:
`(sin x - siny)/(cos x + cos y)= tan (x -y)/2`
Prove the following:
`(sin x + sin 3x)/(cos x + cos 3x) = tan 2x`
Prove the following:
`(cos 4x + cos 3x + cos 2x)/(sin 4x + sin 3x + sin 2x) = cot 3x`
Prove the following:
cos 4x = 1 – 8sin2 x cos2 x
Prove that: sin 3x + sin 2x – sin x = 4sin x `cos x/2 cos (3x)/2`
If \[\sin A = \frac{3}{5}, \cos B = - \frac{12}{13}\], where A and B both lie in second quadrant, find the value of sin (A + B).
Evaluate the following:
cos 47° cos 13° − sin 47° sin 13°
Evaluate the following:
sin 36° cos 9° + cos 36° sin 9°
Prove that:
\[\frac{7\pi}{12} + \cos\frac{\pi}{12} = \sin\frac{5\pi}{12} - \sin\frac{\pi}{12}\]
Prove that:
\[\cos^2 45^\circ - \sin^2 15^\circ = \frac{\sqrt{3}}{4}\]
Prove that:
Prove that:
tan 36° + tan 9° + tan 36° tan 9° = 1
Prove that:
tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x
If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B.
If cos A + sin B = m and sin A + cos B = n, prove that 2 sin (A + B) = m2 + n2 − 2.
If sin α + sin β = a and cos α + cos β = b, show that
Prove that:
\[\frac{1}{\sin \left( x - a \right) \sin \left( x - b \right)} = \frac{\cot \left( x - a \right) - \cot \left( x - b \right)}{\sin \left( a - b \right)}\]
If angle \[\theta\] is divided into two parts such that the tangents of one part is \[\lambda\] times the tangent of other, and \[\phi\] is their difference, then show that\[\sin\theta = \frac{\lambda + 1}{\lambda - 1}\sin\phi\]
Find the maximum and minimum values of each of the following trigonometrical expression:
\[5 \cos x + 3 \sin \left( \frac{\pi}{6} - x \right) + 4\]
Reduce each of the following expressions to the sine and cosine of a single expression:
\[\sqrt{3} \sin x - \cos x\]
Write the maximum value of 12 sin x − 9 sin2 x.
If tan (A + B) = p and tan (A − B) = q, then write the value of tan 2B.
If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).
If tan \[\alpha = \frac{1}{1 + 2^{- x}}\] and \[\tan \beta = \frac{1}{1 + 2^{x + 1}}\] then write the value of α + β lying in the interval \[\left( 0, \frac{\pi}{2} \right)\]
If A + B + C = π, then sec A (cos B cos C − sin B sin C) is equal to
The value of \[\cos^2 \left( \frac{\pi}{6} + x \right) - \sin^2 \left( \frac{\pi}{6} - x \right)\] is
The value of cos (36° − A) cos (36° + A) + cos (54° + A) cos (54° − A) is
If tan (A − B) = 1 and sec (A + B) = \[\frac{2}{\sqrt{3}}\], the smallest positive value of B is
The maximum value of \[\sin^2 \left( \frac{2\pi}{3} + x \right) + \sin^2 \left( \frac{2\pi}{3} - x \right)\] is
If angle θ is divided into two parts such that the tangent of one part is k times the tangent of other, and Φ is their difference, then show that sin θ = `(k + 1)/(k - 1)` sin Φ
If α + β = `pi/4`, then the value of (1 + tan α)(1 + tan β) is ______.
If tanθ = `a/b`, then bcos2θ + asin2θ is equal to ______.
If sinx + cosx = a, then sin6x + cos6x = ______.
