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

Answer: 50

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