Question

If $A=\pi /6$ and $B=\pi /3$, then which of the following is/are correct?
1. $\sin A+\sin B=\cos A+\cos B$
2. $\tan A+\tan B=\cot A+\cot B$
Select the correct answer using the code given below

• A. $1$ only
• B. $2$ only
• C. Both $1$ and $2$
• D. Neither $1$ and $2$

Answer 1

Answer: (C)

Both $1$ and $2$

For $\left(1\right)LHS=\sin \frac{\pi }{6}+\sin \frac{\pi }{3}=\frac{1}{2}+\frac{\sqrt{3}}{2}$
$RHS=\cos \frac{\pi }{6}+\cos \frac{\pi }{3}=\frac{\sqrt{3}}{2}+\frac{1}{2}$
$LHS=RHS$ Hence true.
For $\left(2\right)=LHS=\tan \frac{\pi }{6}+\tan \frac{\pi }{3}=\frac{1}{\sqrt{3}}+\sqrt{3}$
$RHS=cot\frac{\pi }{6}+\tan \frac{\pi }{3}=\sqrt{3}+\frac{1}{\sqrt{3}}$
$LHS=RHS$ Hence true.
Hence both $\left(1\right)$ and $\left(2\right)$ are correct.