Find x if sec²x+tan²x=3 - problem and solution

Find $x$ if $\sec^2{x}+\tan^2{x}=3$

From our trigonometric identities

$\sec^2{x}=\frac{1}{\cos^2{x}}$

$\tan^2{x}=\frac{\sin^2{x}}{\cos^2{x}}$

Now, $\frac{1}{\cos^2{x}}+\frac{\sin^2{x}}{\cos^2{x}}=3$

$\frac{1+\sin^2{x}}{\cos^{x}}=3$

$\Rightarrow{1+\sin^2{x}}=3\cos^2{x}$

$3\cos^2{x}-\sin^2{x}=1$

$3\cos^2{x}-(1-\cos^2{x})=1$

$3\cos^2{x}-1+\cos^2{x}=1$

$3\cos^2{x}+\cos^2{x}=2$

$4\cos^2{x}=2$

Divide through by 4

$\cos^2{x}=\frac{1}{2}$

$\sqrt{\cos^2{x}}=\sqrt{\frac{1}{2}}$

$\cos{x}=\sqrt{\frac{1}{2}}$

$x=\cos^{-1}(\sqrt{\frac{1}{2}})$

$x=45°$