Advertisements
Advertisements
Question
Advertisements
Solution
\[LHS = \sin^3 x + \sin^3 \left( \frac{2\pi}{3} + x \right) + \sin^3 \left( \frac{4\pi}{3} + x \right)\]
\[ = \frac{3\text{ sin } x - \sin3x}{4} + \frac{3\sin\left( \frac{2\pi}{3} + x \right) - \sin3\left( \frac{2\pi}{3} + x \right)}{4} + \frac{3\sin\left( \frac{4\pi}{3} + x \right) - \sin3\left( \frac{4\pi}{3} + x \right)}{4} \]
\[ \left[ \sin^3 \theta = \frac{3sin\theta - \sin3\theta}{4} \right]\]
\[ = \frac{3\text{ sin } x - \sin3x}{4} + \frac{3\sin\left\{ \pi - \left( \frac{2\pi}{3} + x \right) \right\} - \sin\left( 2\pi + 3x \right)}{4} + \frac{3\sin\left\{ \pi + \left( \frac{\pi}{3} + x \right) \right\} - \sin\left( 4\pi + 3x \right)}{4}\]
\[ = \frac{1}{4}\left[ \left( 3\text{ sin } x - \sin3x \right) + \left\{ 3\sin\left( \frac{\pi}{3} - x \right) - \sin3x \right\} - \left\{ 3\sin\left( \frac{\pi}{3} + x \right) + \sin3x \right\} \right]\]
\[ = \frac{1}{4}\left[ 3\text{ sin } x - \sin3x + 3\sin\left( \frac{\pi}{3} - x \right) - 3\sin\left( \frac{\pi}{3} + x \right) - \sin3x - \sin3x \right]\]
\[= \frac{1}{4}\left[ 3\text{ sin } x - 3\sin3x + 3\left\{ \sin\left( \frac{\pi}{3} - x \right) - \sin\left( \frac{\pi}{3} + x \right) \right\} \right]\]
\[ = \frac{1}{4}\left[ 3\text{ sin } x - 3\sin3x + 3\left\{ 2\cos\frac{\frac{\pi}{3} - x + \frac{\pi}{3} + x}{2}\sin\frac{\frac{\pi}{3} - x - \frac{\pi}{3} - x}{2} \right\} \right]\]
\[ \left[ \because sinC - sinD = 2\cos\frac{C + D}{2}\sin\frac{C - D}{2} \right]\]
\[ = \frac{1}{4}\left[ 3\text{ sin } x - 3\sin3x + 6\cos\frac{\pi}{3}\sin\left( - x \right) \right]\]
\[ = \frac{1}{4}\left[ 3\text{ sin } x - 3\text{ sin } 3x - 3\text{ sin } x \right]\]
\[ = - \frac{3}{4}\text{ sin } x\]
\[ = RHS\]
\[\text{ Hence proved } .\]
APPEARS IN
RELATED QUESTIONS
Prove that: \[\frac{\sin 2x}{1 + \cos 2x} = \tan x\]
Prove that: \[\frac{\cos x}{1 - \sin x} = \tan \left( \frac{\pi}{4} + \frac{x}{2} \right)\]
Prove that: \[\cos^2 \frac{\pi}{8} + \cos^2 \frac{3\pi}{8} + \cos^2 \frac{5\pi}{8} + \cos^2 \frac{7\pi}{8} = 2\]
Prove that: \[\left( \cos \alpha + \cos \beta^2 \right) + \left( \sin \alpha + \sin \beta \right)^2 = 4 \cos^2 \left( \frac{\alpha - \beta}{2} \right)\]
Prove that: \[\left( \sin 3x + \sin x \right) \sin x + \left( \cos 3x - \cos x \right) \cos x = 0\]
Prove that: \[\cos^2 \left( \frac{\pi}{4} - x \right) - \sin^2 \left( \frac{\pi}{4} - x \right) = \sin 2x\]
Prove that \[\sin 3x + \sin 2x - \sin x = 4 \sin x \cos\frac{x}{2} \cos\frac{3x}{2}\]
If \[\cos x = - \frac{3}{5}\] and x lies in IInd quadrant, find the values of sin 2x and \[\sin\frac{x}{2}\] .
Prove that: \[\cos 7° \cos 14° \cos 28° \cos 56°= \frac{\sin 68°}{16 \cos 83°}\]
Prove that: \[\cos\frac{2\pi}{15} \cos\frac{4\pi}{15} \cos \frac{8\pi}{15} \cos \frac{16\pi}{15} = \frac{1}{16}\]
If \[a \cos2x + b \sin2x = c\] has α and β as its roots, then prove that
(ii) \[\tan\alpha \tan\beta = \frac{c - a}{c + a}\]
If \[a \cos2x + b \sin2x = c\] has α and β as its roots, then prove that
(iii)\[\tan\left( \alpha + \beta \right) = \frac{b}{a}\]
Prove that: \[\cos^3 x \sin 3x + \sin^3 x \cos 3x = \frac{3}{4} \sin 4x\]
Prove that `tan x + tan (π/3 + x) - tan(π/3 - x) = 3tan 3x`
\[\tan x + \tan\left( \frac{\pi}{3} + x \right) - \tan\left( \frac{\pi}{3} - x \right) = 3 \tan 3x\]
Prove that: \[\sin^2 42° - \cos^2 78 = \frac{\sqrt{5} + 1}{8}\]
Prove that: \[\cos 36° \cos 42° \cos 60° \cos 78° = \frac{1}{16}\]
Prove that: \[\sin\frac{\pi}{5}\sin\frac{2\pi}{5}\sin\frac{3\pi}{5}\sin\frac{4\pi}{5} = \frac{5}{16}\]
If \[\cos 4x = 1 + k \sin^2 x \cos^2 x\] , then write the value of k.
If \[\frac{\pi}{2} < x < \frac{3\pi}{2}\] , then write the value of \[\sqrt{\frac{1 + \cos 2x}{2}}\]
In a right angled triangle ABC, write the value of sin2 A + Sin2 B + Sin2 C.
If \[\text{ tan } A = \frac{1 - \text{ cos } B}{\text{ sin } B}\]
, then find the value of tan2A.
If in a \[∆ ABC, \tan A + \tan B + \tan C = 0\], then
If \[\cos x = \frac{1}{2} \left( a + \frac{1}{a} \right),\] and \[\cos 3 x = \lambda \left( a^3 + \frac{1}{a^3} \right)\] then \[\lambda =\]
If \[2 \tan \alpha = 3 \tan \beta, \text{ then } \tan \left( \alpha - \beta \right) =\]
If \[5 \sin \alpha = 3 \sin \left( \alpha + 2 \beta \right) \neq 0\] , then \[\tan \left( \alpha + \beta \right)\] is equal to
The value of \[\frac{\cos 3x}{2 \cos 2x - 1}\] is equal to
If \[\tan \frac{x}{2} = \frac{\sqrt{1 - e}}{1 + e} \tan \frac{\alpha}{2}\] , then \[\cos \alpha =\]
If \[\tan x = t\] then \[\tan 2x + \sec 2x =\]
If A = cos2θ + sin4θ for all values of θ, then prove that `3/4` ≤ A ≤ 1.
The value of sin 20° sin 40° sin 60° sin 80° is ______.
The value of `cos pi/5 cos (2pi)/5 cos (4pi)/5 cos (8pi)/5` is ______.
If tanθ + sinθ = m and tanθ – sinθ = n, then prove that m2 – n2 = 4sinθ tanθ
[Hint: m + n = 2tanθ, m – n = 2sinθ, then use m2 – n2 = (m + n)(m – n)]
If θ lies in the first quadrant and cosθ = `8/17`, then find the value of cos(30° + θ) + cos(45° – θ) + cos(120° – θ).
If tanθ = `1/2` and tanΦ = `1/3`, then the value of θ + Φ is ______.
The value of `(1 - tan^2 15^circ)/(1 + tan^2 15^circ)` is ______.
