Question

If $L$ denotes the least value of the expression $y=9 \sec ^{2}$ $x+16 \text{cosec}^{2} x$ and $M$ denotes the maximum value of the expression $y=\sin ^{2} x+8 \cos x-7$, find the value of $(L+M)$.

Answer 1

$L=9+16+9 \tan ^{2} x+16 \cot ^{2} x$
$=25+(3 \tan x-4 \cot x)^{2}+24$
$\therefore L=49$ when $\tan ^{2} x=4 / 3$
$M=1-\cos ^{2} x+8 \cos x-7$
$=-\left[\cos ^{2} x-8 \cos x+6\right]$
$=-\left[(\cos x-4)^{2}-10\right]$
$=10-(4-\cos x)^{2}$
$\therefore M=10-9=1$
$\therefore L+M=49+1=50$