Question : If $x+\frac{1}{x}=2 \cos \theta$, then $x^3+\frac{1}{x^3}=?$
Option 1: $2 \cos 2θ$
Option 2: $\cos 3θ$
Option 3: $2 \cos 3θ$
Option 4: $\cos 2θ$
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Correct Answer: $2 \cos 3θ$
Solution : Given: $x+\frac{1}{x}=2 \cos \theta$. Cubing both sides, we get: $⇒\left(x^3+\frac{1}{x^3}\right) + 3\left(x+\frac{1}{x}\right) = 8\cos^3 \theta$ Putting the values, we get: $⇒x^3+\frac{1}{x^3} = 8\cos^3 \theta- 3(2 \cos \theta)$ $⇒x^3+\frac{1}{x^3}= 8\cos^3 \theta- 6 \cos \theta$ $⇒x^3+\frac{1}{x^3}=2(4\cos^3 \theta- 3 \cos \theta)$ $⇒x^3+\frac{1}{x^3}=2 \cos 3 \theta$ Hence, the correct answer is $2 \cos 3 \theta$.
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Question : If $x\sin^{3}\theta +y\cos^{3}\theta=\sin\theta\cos\theta$ and $x\sin\theta-y\cos\theta=0$, then the value of $\left ( x^{2}+y^{2} \right )$ equals:
Question : What is $\tan \frac{\theta}{2}$?
Question : If $\cos \theta=\frac{\sqrt{3}}{2}$, then $\tan ^2 \theta \cos ^2 \theta=?$
Question : If $\tan\theta=1$, then the value of $\frac{8\sin\theta\:+\:5\cos\theta}{\sin^{3}\theta\:–\:2\cos^{3}\theta\:+\:7\cos\theta}$ is:
Question : If $\tan \theta=\frac{4}{3}$, then the value of $\frac{3\sin \theta+ 2\cos \theta}{3\sin \theta – 2 \cos \theta}$ is:
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