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:
Option 1: $2$
Option 2: $2\frac{1}{2}$
Option 3: $3$
Option 4: $\frac{4}{5}$
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Correct Answer: $2$
Solution :
Given: $\tan\theta=1$
$\tan\theta=\tan45^{\circ}$
⇒ $\theta=45^{\circ}$
Putting the value of $\theta$ in the given expression,
$= \frac{8\sin45^{\circ}\:+\:5\cos45^{\circ}}{\sin^{3}45^{\circ}\:-\:2\cos^{3}45^{\circ}\:+\:7\cos45^{\circ}}$
$= \frac{8×\frac{1}{\sqrt{2}}\:+\:5×\frac{1}{\sqrt{2}}}{(\frac{1}{\sqrt{2}})^{3}\:-\:2×(\frac{1}{\sqrt{2}})^{3}\:+\:7×\frac{1}{\sqrt{2}}}$
$= \frac{(8\:+\:5)×\frac{1}{\sqrt{2}}}{(\frac{1}{2\sqrt{2}})\:-\:2×(\frac{1}{2\sqrt{2}})\:+\:(\frac{7}{\sqrt{2}} \times \frac{2}{2})}$
$= \frac{\frac{13}{\sqrt{2}}}{\frac{1}{2\sqrt{2}}\:×\:(1\:-\:2\:+\;14)}$
$= \frac{\frac{13}{\sqrt{2}}}{\frac{13}{2\sqrt{2}}}$
$= 2$
Hence, the correct answer is 2.
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