Question

tan^{2}(x) + sec^{2}(x) = 2

Explanation:

Step1: Use trigonometric identity

We know that \(\sec^{2}(x)=1 + \tan^{2}(x)\). Substitute this into the given equation \(\tan^{2}(x)+\sec^{2}(x)=2\), we get \(\tan^{2}(x)+(1 + \tan^{2}(x))=2\).

Step2: Simplify the equation

Expand the left - hand side: \(2\tan^{2}(x)+1 = 2\). Then subtract 1 from both sides: \(2\tan^{2}(x)=1\). Divide both sides by 2: \(\tan^{2}(x)=\frac{1}{2}\).

Step3: Solve for \(\tan(x)\)

Take the square root of both sides: \(\tan(x)=\pm\frac{\sqrt{2}}{2}\).

Step4: Find \(x\)

If \(\tan(x)=\frac{\sqrt{2}}{2}\), then \(x = n\pi+\arctan(\frac{\sqrt{2}}{2})\), \(n\in\mathbb{Z}\). If \(\tan(x)=-\frac{\sqrt{2}}{2}\), then \(x=n\pi-\arctan(\frac{\sqrt{2}}{2})\), \(n\in\mathbb{Z}\).

Answer:

\(x = n\pi\pm\arctan(\frac{\sqrt{2}}{2})\), \(n\in\mathbb{Z}\)