Question
Determine the value of \sin^2 20\degree +\cos^2 50\degree +\sin 20\degree \cos 50\degree
Determine the value of \sin^2 20\degree +\cos^2 50\degree +\sin 20\degree \cos 50\degree
Using power reducing, product to sum and sum to product identities
\sin^2\alpha = \dfrac{1-\cos (2\alpha)}{2}
\cos^2\alpha = \dfrac{1+\cos (2\alpha)}{2}
\sin \alpha\cos \beta = \dfrac{1}{2} [\sin(\alpha+\beta ) +\sin(\alpha - \beta) ]
\cos \alpha - \cos \beta = -2\sin\dfrac{\alpha +\beta }{2} \sin\dfrac{\alpha -\beta }{2}
\sin^2 20\degree +\cos^2 50\degree +\sin 20\degree \cos 50\degree
= \dfrac{1-\cos 40\degree }{2}+ \dfrac{1+\cos 100\degree }{2} +\dfrac{1}{2}[\sin 70\degree +\sin( -30\degree ) ]
= 1-\dfrac{1}{4}+\dfrac{\cos 100\degree-\cos 40\degree}{2} +\dfrac{1}{2}\sin 70\degree
=\dfrac{3}{4}- \sin 70\degree \sin 30\degree +\dfrac{1}{2}\sin 70\degree
=\dfrac{3}{4}