Question

( begin{aligned} F(8) &=sin ^{2} theta+cos 2 theta &=sin 4 theta+1-sin ^{2} Q &=sin theta-sin ^{2} theta+1 &=left(sin ^{2} thetaright)^{2}-2 cdot frac{1}{2} sin ^{2} theta+left(frac{1}{2}right)^{2}-frac{1}{4}+1 &=left(sin ^{2} theta-frac{1}{2}right)^{2}+frac{3}{4} &=frac{1}{4} &=frac{1}{2} leq sin ^{2} theta-frac{1}{2} leq 1-frac{1}{2} &left(frac{1}{4}right) leqleft(sin ^{2} theta-frac{1}{2}right)^{2} leq frac{1}{4} frac{1}{4}+frac{3}{4} & leqleft(sin ^{2} theta-frac{1}{2}right)^{2}+frac{3}{4} leq frac{1}{4}+frac{3}{4} & 1 leq f(theta) leq 1 end{aligned} )

# If f(0) = sin4 0 + cos2 0, then range of f(0) is 9

Solution