Question

Solve ${\sin }^{2}x+{\cos }^{2}y=2{\sec }^{2}z$ for x, y and z

• A. $x=(2m+1)\frac{\pi }{2}$, here m is integer
• B. $y=n\pi$ where n is integer
• C. $z=t\pi$, where t is integer
• D. $x=(2m+1)\frac{\pi }{4}$, here m is integer

Answer 1

Answer: (A), (B), (C)

The correct options are
A $x=(2m+1)\frac{\pi }{2}$, here m is integer
B $z=t\pi$, where t is integer
C $y=n\pi$ where n is integer

L.H.S. $={\sin }^{2}x+{\cos }^{2}y\le 2$ [$\because {\sin }^{2}x\le$ and ${\cos }^{2}y\le 1$]

R.H.S. $=2{\sec }^{2}z\ge 2$

Hence, L.H.S.=R.H.S. only when ${\sin }^{2}x=1$, ${\cos }^{2}y=1$ and $2{\sec }^{2}z=2$. Thus,

${\cos }^{2}x=0$, ${\sin }^{2}y=0$, ${\cos }^{2}z=1$

$\Rightarrow$ $\cos x=0$, $\sin y=0$, $\sin z=0$

$x=(2m+1)\frac{\pi }{2}$, $y=n\pi$ and $z=t\pi$, where m, n and t are integers