Question

If $A={30}^{\circ }$ and $B={60}^{\circ }$, then which of the following is/are correct?
I. $\sin A+\sin B=\cos A+\cos B$
II. $\tan A+\tan B=\cot A+\cot B$
Select the correct answer using the code given below:

• A. Only I
• B. Only II
• C. Both I and II
• D. Neither I nor II

Answer 1

The correct option is A Only I

Given that, $A=30$ $B=60$

(I) $\sin A+\sin B=\cos A+\cos B$

Consider,

$\sin A+\sin B$ $=\sin 30+\sin 60$

$=\frac{1}{2}+\frac{\sqrt{3}}{2}=\frac{1+\sqrt{3}}{2}$

Consider, $\cos A+\cos B$ $\cos 30+\cos 60$

$=\frac{\sqrt{3}}{2}+\frac{1}{2}$

$=\frac{\sqrt{3}+1}{2}$

$\therefore \sin A+\sin B=\cos A+\cos B$ (true)

(II) $\tan A+\tan B=\cot A+\cot B$

$\tan A+\tan B=\tan 30+\tan 60$

$=\frac{1}{\sqrt{3}}+\sqrt{3}=\frac{1+3}{\sqrt{3}}$

$=\frac{4}{\sqrt{3}}$

$\cot A+\cot B=\cot 30+\cot 60$

$=\sqrt{3}+\frac{1}{\sqrt{3}}$

$=\frac{3+1}{\sqrt{3}}=\frac{4}{\sqrt{3}}$

$\tan A+\tan B=\cot A+\cot B$ (true)