Question

${\sin }^2{5}^{\circ }+{\sin }^2{10}^{\circ }+{\sin }^2{15}^{\circ }+....+{\sin }^2{180}^{\circ }=$

• A. $9$
• B. $12$
• C. $18$
• D. $24$

Answer 1

Answer: (C)

$18$

${\sin }^2{5}^o+{\sin }^2{10}^o..........{\sin }^2{90}^o+{\sin }^2{95}^o+..............{\sin }^2{180}^o$

$={\sin }^2{5}^o+{\sin }^2{10}^o..........{\sin }^2{90}^o+{\sin }^2({180}^o-{95}^o)+..............{\sin }^2({180}^o-{180}^o)$

$={\sin }^2{5}^o+{\sin }^2{10}^o..........{\sin }^2{90}^o+{\sin }^2\left({85}^o\right)+..............{\sin }^2\left(0\right)$

$=2({\sin }^2{5}^o+{\sin }^2{10}^o.........+{\sin }^2{85}^o)+{\sin }^2{90}^o$

$=2({\sin }^2{5}^o+{\sin }^2{10}^o.........+{\sin }^2{45}^o+{\cos }^2{40}^o+...........{\cos }^2{5}^o)+{\sin }^2{90}^o$

$=2({\sin }^2{5}^o+{\cos }^2{5}^o+..............{\sin }^2{40}^o+{\cos }^2{40}^o+{\sin }^2{45}^o)+{\sin }^2{90}^o$

$=2(1+1+1+1+1+1+1+1+\frac{1}{2})+1$

$=18$

Hence, the answer is $18$.