The number of values of x ∈ (–π, π) satisfying 2 tan

Question:

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

Solution:

For x∊ (−π, π)

2tan2x = sec2x

i. e $\frac{2 \sin ^{2} x}{\cos ^{2} x}=\frac{1}{\cos ^{2} x} \quad$ i. e $x \notin(2 n+1) \frac{\pi}{2}$

i. e $\sin ^{2} x=\frac{1}{2}$

i. e $\sin x=\pm \frac{1}{\sqrt{2}}$

i. e $x=\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}$

∴ Number of value of x in (−π, π)

Satisfying 2tan2x = sec2x is 4.

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