Advertisements
Advertisements
Question
Prove the following:
`(sin x - sin 3x)/(sin^2 x - cos^2 x) = 2sin x`
Advertisements
Solution
We have, L.H.S. = `(sin x - sin 3x)/(sin^2 x - cos^2 x)`
= `(2sin ((x - 3x)/2) cos ((x + 3x)/2))/(-cos^2x - sin^2x)`
= `(2sin (-x) cos (2x))/(-cos2x)`
= `(- 2sin x cos(2x))/(-cos2x)` [∵ cos2x - sin2x = cos 2x]
= `(-2 sinx)/-1` [∵ cos2x - sin2x = cos 2x]
= 2 sin x = R.H.S.
APPEARS IN
RELATED QUESTIONS
Prove that `2 sin^2 pi/6 + cosec^2 (7pi)/6 cos^2 pi/3 = 3/2`
Prove that `cot^2 pi/6 + cosec (5pi)/6 + 3 tan^2 pi/6 = 6`
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:
cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1
Prove the following:
`tan 4x = (4tan x(1 - tan^2 x))/(1 - 6tan^2 x + tan^4 x)`
Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x
If \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:
cos (A − B)
If \[\sin A = \frac{1}{2}, \cos B = \frac{\sqrt{3}}{2}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
tan (A - B)
Evaluate the following:
cos 47° cos 13° − sin 47° sin 13°
If \[\cos A = - \frac{12}{13}\text{ and }\cot B = \frac{24}{7}\], where A lies in the second quadrant and B in the third quadrant, find the values of the following:
sin (A + B)
If \[\cos A = - \frac{12}{13}\text{ and }\cot B = \frac{24}{7}\], where A lies in the second quadrant and B in the third quadrant, find the values of the following:
tan (A + B)
Prove that:
\[\frac{7\pi}{12} + \cos\frac{\pi}{12} = \sin\frac{5\pi}{12} - \sin\frac{\pi}{12}\]
Prove that
Prove that
Prove that:
Prove that:
\[\tan\frac{\pi}{12} + \tan\frac{\pi}{6} + \tan\frac{\pi}{12}\tan\frac{\pi}{6} = 1\]
Prove that:
tan 13x − tan 9x − tan 4x = tan 13x tan 9x tan 4x
Prove that:
\[\frac{\tan^2 2x - \tan^2 x}{1 - \tan^2 2x \tan^2 x} = \tan 3x \tan x\]
If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B.
If α and β are two solutions of the equation a tan x + b sec x = c, then find the values of sin (α + β) and cos (α + β).
Reduce each of the following expressions to the sine and cosine of a single expression:
\[\sqrt{3} \sin x - \cos x\]
Show that sin 100° − sin 10° is positive.
If 12 sin x − 9sin2 x attains its maximum value at x = α, then write the value of sin α.
If \[\frac{\cos \left( x - y \right)}{\cos \left( x + y \right)} = \frac{m}{n}\] then write the value of tan x tan y.
The value of \[\sin^2 \frac{5\pi}{12} - \sin^2 \frac{\pi}{12}\]
If \[\tan A = \frac{a}{a + 1}\text{ and } \tan B = \frac{1}{2a + 1}\]
Express the following as the sum or difference of sines and cosines:
2 sin 3x cos x
Express the following as the sum or difference of sines and cosines:
2 cos 7x cos 3x
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 3 tan (θ – 15°) = tan (θ + 15°), 0° < θ < 90°, then θ = ______.
The value of tan 75° - cot 75° is equal to ______.
The value of sin(45° + θ) - cos(45° - θ) is ______.
If tanA = `1/2`, tanB = `1/3`, then tan(2A + B) is equal to ______.
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 |sinx – cosx| = ______.
The maximum distance of a point on the graph of the function y = `sqrt(3)` sinx + cosx from x-axis is ______.
