What is the general solution for 2sec^2(x)=3-tan(x) ?

The general solution for 2sec^2(x)=3-tan(x) is x=0.46364…+pin,x=(3pi)/4+pin

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2sec^2(x)=3-tan(x)

2 sec^{2} (x) = 3 - tan (x)

x = 0.46364 \dots + πn, x = \frac{3 π}{4} + πn

Solution steps

2 sec^{2} (x) = 3 - tan (x)

Subtract

3 - tan (x)

from both sides

2 sec^{2} (x) - 3 + tan (x) = 0

Rewrite using trig identities

- 1 + tan (x) + 2 tan^{2} (x) = 0

Solve by substitution

tan (x) = \frac{1}{2}, tan (x) = - 1

tan (x) = \frac{1}{2} : x = arctan (\frac{1}{2}) + πn

tan (x) = - 1 : x = \frac{3 π}{4} + πn

Combine all the solutions

x = arctan (\frac{1}{2}) + πn, x = \frac{3 π}{4} + πn

Show solutions in decimal form

x = 0.46364 \dots + πn, x = \frac{3 π}{4} + πn

2 sin^{2} (θ) - sin (θ) = 1

2 sin (- 3 x - \frac{π}{4}) = 2

4 sin^{3} (θ) - 3 sin (θ) = 0

\frac{sec ^{2} ( θ ) - 1}{1 - cos ^{2} ( θ )} - 1 = cot^{2} (θ)

cos^{2} (x) + cos (x) = sin^{2} (x)

What is the general solution for 2sec^2(x)=3-tan(x) ?
The general solution for 2sec^2(x)=3-tan(x) is x=0.46364…+pin,x=(3pi)/4+pin