Question

If $a=30$, then $\frac{{\cos }^{3}a-\cos 3a}{\cos a}+\frac{{\sin }^{3}a+\sin 3a}{\sin a}=$

Answer 1

Compute the required value.

Given: $\frac{{\cos }^{3}a-\cos 3a}{\cos a}+\frac{{\sin }^{3}a+\sin 3a}{\sin a}$ and $a=30$

$=\frac{{\cos }^{3}30°-\cos 90°}{\cos 30°}+\frac{{\sin }^{3}30°+\sin 90°}{\sin 30°}$

$=$$\frac{{\left({\displaystyle \frac{\sqrt{3}}{2}}\right)}^3}{{\displaystyle \frac{\sqrt{3}}{2}}}+\frac{{\left({\displaystyle \frac{1}{2}}\right)}^3+1}{{\displaystyle \frac{1}{2}}}$ $$\begin{gathered} \left[\cos 30°=\frac{\sqrt{3}}{2},\cos 90°=0 \\ \sin 30°=\frac{1}{2},\sin 90°=1\right] \end{gathered}$$

$=\frac{3}{4}+\frac{9}{4}$

$=\frac{12}{4}$

$=3$

Hence, the value of $\frac{{\cos }^{3}a-\cos 3a}{\cos a}+\frac{{\sin }^{3}a+\sin 3a}{\sin a}$ is $3$ when $a=30$