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JEE MATHEMATICS MCQ: MULTIPLE AND SUBMULTIPLE ANGLES

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For any angle θ\theta, the expression 2cos⁡  8θ+12cos⁡  θ+1\frac{{2\cos \,\,8\theta + 1}}{{2\cos \,\,\theta + 1}} is equal to
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Correct option: C
(2  cos⁡  θ−1)  (2  cos⁡  2θ−1)  (2  cos⁡  4θ−1)\left( {2\,\,\cos \,\,\theta - 1} \right)\,\,\left( {2\,\,\cos \,\,2\theta - 1} \right)\,\,\left( {2\,\,\cos \,\,4\theta - 1} \right)
Option C ((2 cos⁡ θ−1) (2 cos⁡ 2θ−1) (2 cos⁡ 4θ−1)\left( {2\,\cos \,\theta - 1} \right)\,\left( {2\,\cos \,2\theta - 1} \right)\,\left( {2\,\cos \,4\theta - 1} \right)) is correct because it is obtained by repeatedly applying the double angle formula for cosine and simplifying the expression.

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If 1+cos⁡  x=k1 + \cos \,\,x = k, where xx is acute, then sin⁡  x2\sin \,\,\frac{x}{2} is The value of sin⁡12∘⋅sin⁡48∘⋅sin⁡54∘=\sin 12^\circ \cdot \sin 48^\circ \cdot \sin 54^\circ = If α,    β(α≠β)\alpha ,\;\,\,\beta \left( {\alpha \ne \beta } \right) satisfies the equation a a  cos⁡  θ+b  sin⁡  θ=ca\,\,\cos \,\,\theta + b\,\,\sin \,\,\theta = c, then the value of tan⁡  (α+β2)\tan \,\,\left( {\frac{{\alpha + \beta }}{2}} \right) is

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cos⁡210∘−cos⁡10∘cos⁡50∘+cos⁡250∘\cos ^{2} 10^{\circ}-\cos 10^{\circ} \cos 50^{\circ}+\cos ^{2} 50^{\circ} is : In a △PQR,∠R=π2\triangle P Q R, \angle R=\frac{\pi}{2}. If tan⁡(P2)\tan \left(\frac{P}{2}\right) and tan⁡(Q2)\tan \left(\frac{Q}{2}\right) are the roots of ax2+bx+c=0,a≠0a x^2+b x+c=0, a \neq 0, then
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