Ravi Kant Kumar

Prove that sin³x + sin³(2π/3 + x) + sin³(4π/3 + x) = −3/4 sin3x Prove that: \[ \sin^3 x+\sin^3\left(\frac{2\pi}{3}+x\right) +\sin^3\left(\frac{4\pi}{3}+x\right) = -\frac{3}{4}\sin 3x \] Solution Using the identity \[ \sin^3\theta = \frac{3\sin\theta-\sin3\theta}{4} \] we get \[ \sin^3 x = \frac{3\sin x-\sin3x}{4} \] \[ \sin^3\left(\frac{2\pi}{3}+x\right) = \frac{ 3\sin\left(\frac{2\pi}{3}+x\right) – \sin\left(2\pi+3x\right) }{4} \] \[ \sin^3\left(\frac{4\pi}{3}+x\right) = \frac{

Prove that sin5x = 5cos⁴x sinx − 10cos²x sin³x + sin⁵x Prove that: \[ \sin 5x = 5\cos^4 x \sin x -10\cos^2 x \sin^3 x +\sin^5 x \] Solution We know that \[ \sin 5x = \sin(4x+x) \] Using the formula \[ \sin(A+B) = \sin A\cos B+\cos A\sin B \] we get \[ \sin 5x

Prove that cotx + cot(π/3 + x) + cot(2π/3 + x) = 3 cot3x Prove that: \[ \cot x+\cot\left(\frac{\pi}{3}+x\right) +\cot\left(\frac{2\pi}{3}+x\right) = 3\cot 3x \] Solution Let \[ t=\cot x \] Using \[ \cot(A+B) = \frac{\cot A\cot B-1}{\cot A+\cot B} \] and \[ \cot\frac{\pi}{3}=\frac{1}{\sqrt{3}} \] \[ \cot\frac{2\pi}{3}=-\frac{1}{\sqrt{3}} \] we get \[ \cot\left(\frac{\pi}{3}+x\right) = \frac{\frac{t}{\sqrt{3}}-1}{\frac{1}{\sqrt{3}}+t} = \frac{t-\sqrt{3}}{1+\sqrt{3}t}

Prove that cotx + cot(π/3 + x) − cot(π/3 − x) = 3 cot3x Prove that: \[ \cot x+\cot\left(\frac{\pi}{3}+x\right) -\cot\left(\frac{\pi}{3}-x\right) = 3\cot 3x \] Solution Let \[ \cot x=t \] Using \[ \cot(A+B) = \frac{\cot A\cot B-1}{\cot A+\cot B} \] \[ \cot(A-B) = \frac{\cot A\cot B+1}{\cot B-\cot A} \] and \[ \cot\frac{\pi}{3}=\frac{1}{\sqrt{3}} \] we get

Prove that tanx + tan(π/3 + x) − tan(π/3 − x) = 3 tan3x Prove that: \[ \tan x+\tan\left(\frac{\pi}{3}+x\right) -\tan\left(\frac{\pi}{3}-x\right) = 3\tan 3x \] Solution Let \[ \tan x=t \] Using the formula \[ \tan(A\pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B} \] and \[ \tan\frac{\pi}{3}=\sqrt{3} \] we get \[ \tan\left(\frac{\pi}{3}+x\right)

Prove that tanx tan(x + π/3) + tanx tan(x − π/3) + tan(x + π/3) tan(x − π/3) = −3 Prove that: \[ \tan x \tan \left(x+\frac{\pi}{3}\right) +\tan x \tan \left(x-\frac{\pi}{3}\right) +\tan \left(x+\frac{\pi}{3}\right) \tan \left(x-\frac{\pi}{3}\right) =-3 \] Solution Let \[ \tan x = t \] Using the formula \[ \tan(A\pm B) = \frac{\tan A \pm

Prove that cos³x sin3x + sin³x cos3x = 3/4 sin4x Prove that: \[ \cos^3 x \sin 3x + \sin^3 x \cos 3x = \frac{3}{4}\sin 4x \] Solution Start with the left hand side: \[ \cos^3 x \sin 3x + \sin^3 x \cos 3x \] Using the identities \[ \sin 3x = 3\sin x – 4\sin^3

Prove that 4(cos³10° + sin³20°) = 3(cos10° + sin20°) Prove that: \[ 4(\cos^3 10^\circ + \sin^3 20^\circ) = 3(\cos 10^\circ + \sin 20^\circ) \] Solution We use the identity \[ 4a^3 – 3a = \cos 3\theta \quad \text{when } a=\cos\theta \] So, \[ 4\cos^3 10^\circ = 3\cos 10^\circ + \cos 30^\circ \] Also, using \[

Prove that sin 5x = 5 sin x – 20 sin³ x + 16 sin⁵ x Prove that: \( \sin 5x = 5\sin x – 20\sin^3 x + 16\sin^5 x \) Solution We know that \[ \sin 5x = \sin(2x+3x) \] Using the formula \[ \sin(A+B)=\sin A \cos B + \cos A \sin B \]

Value of Trigonometric Functions at Multiples and Submultiples of an angle – Exercise 9.2 Solutions Prove that: sin 5x = 5 sin x – 20 sin³ x + 16 sin⁵ x  Watch Solution Prove that: 4 (cos³ 10° + sin³ 20°) = 3 (cos 10° + sin 20°) Watch Solution Prove that: cos³ x sin 3x