Question

The number of values of x ∈ (–π, π) satisfying 2 tan²x = sec²x is ______________.

Answer 1

For x∊ (−π, π)
2tan²x = sec²x
$$\begin{gathered} \mathrm{i}.\mathrm{e}\frac{2{\sin }^{2}x}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}\mathrm{i}.\mathrm{e}x\notin \left(2n+1\right)\frac{\mathrm{\pi }}{2} \\ \mathrm{i}.\mathrm{e}{\sin }^{2}x=\frac{1}{2} \\ \mathrm{i}.\mathrm{e}\sin x=\pm \frac{1}{\sqrt{2}} \\ \mathrm{i}.\mathrm{e}x=\frac{\mathrm{\pi }}{4},\frac{3\mathrm{\pi }}{4},\frac{5\mathrm{\pi }}{4},\frac{7\mathrm{\pi }}{4} \end{gathered}$$
∴ Number of value of x in (−π, π)
Satisfying 2tan²x = sec²x is 4.