Question

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

[IIT 1964]

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

$1+cos{56}^{\circ }+cos{58}^{\circ }-cos{66}^{\circ }=$
=$2co{s}^2{28}^{\circ }+2sin{62}^{\circ }.sin{4}^{\circ }$
=$2co{s}^2{28}^{\circ }+2cos{28}^{\circ }.sin{4}^{\circ }$
=$2co{s}^2{28}^{\circ }(cos{28}^{\circ }.cos{86}^{\circ })$
=$(2co{s}^2{28}^{\circ }.2cos{57}^{\circ }.cos{29}^{\circ }$
=$4cos{28}^{\circ }cos{29}^{\circ }sin{33}^{\circ }$
Aliter: Apply the condition identify

$cosA+cosB-cosC=-1+4cos\frac{A}{2}cos\frac{B}{2}sin\frac{c}{2}$
$[\therefore {56}^{\circ }+{58}^{\circ }+{66}^{\circ }={180}^{\circ }]$
We get the value of required expression equal to $4cos{28}^{\circ }cos{29}^{\circ }sin{33}^{\circ }$.