tan^2(x)+5=3sec^2(x)

Solution

tan^{2} (x) + 5 = 3 sec^{2} (x)

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Solution steps

tan^{2} (x) + 5 = 3 sec^{2} (x)

Subtract

3 sec^{2} (x)

from both sides

tan^{2} (x) + 5 - 3 sec^{2} (x) = 0

Rewrite using trig identities

4 - 2 sec^{2} (x) = 0

Solve by substitution

sec (x) = 2, sec (x) = - 2

sec (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) = - 2 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

tan (x) = \frac{7}{2}

6 cos^{2} (2 x) + 10 cos (4 x) = 12.05718

cos (\frac{θ}{2}) = \frac{1 + \frac{3}{5}}{2}

cos (θ) = \frac{35}{37}

- \frac{π}{2} = 1 - 2 sin (x)