Question

If $A={60}^{\circ }andB={30}^{\circ }$, then verify each of the following:
$\left(i\right)\cos (A-B)=\cos A\cos B+\sin A\sin B$
$\left(ii\right)\cot (A+B)=\frac{\cot A\cot B-1}{\cot A+\cot B}$

• A. $\left(i\right)True$$\left(ii\right)False$
• B. $\left(i\right)False$$\left(ii\right)False$
• C. $\left(i\right)True$$\left(ii\right)True$
• D. $\left(i\right)False$$\left(ii\right)True$

Answer 1

The correct option is C $\left(i\right)True$

$\left(ii\right)True$

$=cos({60}^0-{30}^0)$
$=cos\left({30}^0\right)$
$=\frac{\sqrt{3}}{2}$
$=LHS$
Now
$RHS$
$=cos{60}^{0}cos{30}^0+sin{30}^{0}sin{60}^0$
$=\frac{\sqrt{3}}{4}+\frac{\sqrt{3}}{4}$
$=\frac{\sqrt{3}}{2}$
$=RHS$
Hence verified.
$=cot({60}^0+{30}^0)$
$=cot\left({90}^0\right)$
$=0$
$=LHS$
RHS
$=\frac{cot{60}^0.cot{30}^0-1}{cot{60}^0+cot{30}^0}$
$=\frac{1-1}{\frac{1}{\sqrt{3}}+\sqrt{3}}$
$=0$
$Henceverified.$