On simplifying ^3 A+ 3 A A + ^3 A- 3 A A , we get

On simplifying \frac{\sin ^3 \mathrm{ A}+\sin 3 \mathrm{ A}}{\sin \mathrm{ A}}+\frac{\cos ^3 \mathrm{ A}-\cos 3 \mathrm{ A}}{\cos \mathrm{ A}}$, we get

Options

A $\sin 3 \mathrm{ A}$
B $\cos 3 \mathrm{ A}$
C $\sin \mathrm{A}+\cos \mathrm{A}$
D3

Correct answer

D. 3

Step-by-step solution

\begin{aligned} & \frac{\sin ^3 \mathrm{ A}+\sin 3 \mathrm{ A}}{\sin \mathrm{ A}}+\frac{\cos ^3 \mathrm{ A}-\cos 3 \mathrm{ A}}{\cos \mathrm{ A}} \\ \Rightarrow & \frac{\sin ^3 \mathrm{ A}+3 \sin \mathrm{ A}-4 \sin ^3 \mathrm{ A}}{\sin \mathrm{ A}}+\frac{\cos ^3 \mathrm{ A}-\left[4 \cos ^3 \mathrm{ A}-3 \cos \mathrm{ A}\right]}{\cos \mathrm{A}} \\ \Rightarrow & \frac{3 \sin \mathrm{ A}-3 \sin ^3 \mathrm{ A}}{\sin \mathrm{ A}}+\frac{\left(-3 \cos ^3 \mathrm{ A}+3 \cos \mathrm{ A}\right)}{\cos \mathrm{A}} \\ = & 3-3 \sin ^2 \mathrm{ A}-3 \cos ^2 \mathrm{ A}+3 \\ = & 6-3\left(\cos ^2 \mathrm{ A}+\sin ^2 \mathrm{ A}\right) \\ = & 6-3(1) \\ = & 3\end{aligned}

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