Question

If $x,y\in (0,\frac{\pi }{2})$,then the expression $\sin x+\cos y+{\tan }^{2}y+{\cot }^{2}x+5$ is always greater than

Answer 1

Given expression:$\sin x+\cos y+{\tan }^{2}y+{\cot }^{2}x+5$

$=\sin x+\cos y+{\sec }^{2}y-1+{\csc }^{2}x-1+5$

$=\sin x+\cos y+{\sec }^{2}y+{\csc }^{2}x+3$

Now,$\sin x+\frac{1}{{\sin }^{2}x}\ge \sin x+\frac{1}{\sin x}\ge 2$

and $\cos y+\frac{1}{{\cos }^{2}y}\ge \cos y+\frac{1}{\cos y}\ge 2$

$\therefore$ the given expression is

$\sin x+\cos y+{\sec }^{2}y+{\csc }^{2}x+3$

$=(\sin x+{\csc }^{2}x)+(\cos y+{\sec }^{2}y)+3$

$\ge 2+2+3=7$

$\therefore$ the given expression$\ge 7$