Question

The value of sin2 1° + sin2 5° + sin2 85° + sin2 89° is

Answer 1

Answer:

2

Step-by-step explanation:

$\sin^{2}1 + \sin {}^{2}5 \: + \sin^{2} 85 + \sin^{2}89 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ...(1)$

convert

$\sin{}^{2}85 \: and \: \sin {}^{2} 89$

in cos term

$\sin^{2} 85 = \sin^{2} (90 - 5) \\ \sin^{2}89 = sin^{2} (90 - 1) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: ....(2)$

now use formula

$\sin(90 - \theta) = cos \theta$

from equation (1) and (2)

$\sin^{2}1 + cos^{2} 1 + \sin {}^{2}5 + \cos^{2}5 \\ = 1 + 1 \\ = 2$

in last step use

$cos^2 + sin^2 = 1$