Question

$1+cos{56}^{\circ }+cos{58}^{\circ }-cos{66}^{\circ }=$

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

$1+cos{56}^{\circ }+cos{58}^{\circ }-cos{66}^{\circ }$
$=2co{s}^2{28}^{\circ }+2sin{62}^{\circ }.sin{4}^{\circ }$
$=2co{s}^2{28}^{\circ }+2cos{28}^{\circ }.sin{4}^{\circ }$
$=2cos{28}^{\circ }(cos{28}^{\circ }+cos{86}^{\circ })$
$=2cos{28}^{\circ }.2cos{57}^{\circ }cos{29}^{\circ }$
$=4cos{28}^{\circ }cos{29}^{\circ }sin{33}^{\circ }.$
Aliter: Apply the conditional identity
$cosA+cosB-cosC=-1+4cos\frac{A}{2}cos\frac{B}{2}sin\frac{C}{2}$
$[\because {56}^{\circ }+{58}^{\circ }+{66}^{\circ }={180}^{\circ }]$
We get the value of required expression equal to $=4cos{28}^{\circ }cos{29}^{\circ }sin{33}^{\circ }.$