Given-A-sin-2-cos-4-then-for-all-real-

Question Number 45830 by jashim last updated on 17/Oct/18

Given A = \sin^{2} θ + \cos^{4} θ, then for all

real θ

Commented by maxmathsup by imad last updated on 17/Oct/18

t h e Q i s a b s e n t l e t s u p p s e s i m p l i f i c a t i o n

A = \frac{1 - c o s (2 θ)}{2} + {(\frac{1 + c o s (2 θ)}{2})}^{2} = \frac{1 - c o s (2 θ)}{2} + \frac{1 + 2 c o s (2 θ) + {c o s}^{2} (2 θ)}{4}

= \frac{2 - 2 c o s (2 θ) + 1 + 2 c o s (2 θ) + {c o s}^{2} (2 θ)}{4} = \frac{3 + {c o s}^{2} (2 θ)}{4}

= \frac{3 + \frac{1 + c o s (4 θ)}{2}}{4} = \frac{6 + 1 + c o s (4 θ)}{8} = \frac{7}{8} + \frac{1}{8} c o s (4 θ) .