First slide

Trigonometric equations

Question

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

Moderate

A

$x = (2 m + 1) \frac{π}{2}, m \in Z$
B

$y = n π, n \in Z$
C

$z = t π, t \in Z$
D

All the above

Solution

$L H S = \sin^{2} x + \cos^{2} y \leq 2$ , $\sin^{2} x \leq 1, \cos^{2} y \leq 1$
$R H S = 2 \sec^{2} z \geq 2$
$L H S = R H S o n l y w h e n$ $\sin^{2} x = 1, \cos^{2} y = 1, 2 \sec^{2} z = 2$
$\Rightarrow \cos^{2} x = 0, \sin^{2} y = 0, \cos^{2} z = 1$
$\Rightarrow \cos x = 0, \sin y = 0, \sin z = 0$
$\Rightarrow x = (2 m + 1) \frac{π}{2}, y = n π$ $a n d z = t π, m, n, t \in Z$

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