6tan^2(x)-2cos^2(x)=cos(2x)

Solution

6 tan^{2} (x) - 2 cos^{2} (x) = cos (2 x)

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Solution steps

6 tan^{2} (x) - 2 cos^{2} (x) = cos (2 x)

Subtract

cos (2 x)

from both sides

6 tan^{2} (x) - 2 cos^{2} (x) - cos (2 x) = 0

Express with sin, cos

\frac{- 2 cos ^{4} ( x ) + 6 sin ^{2} ( x ) - cos ( 2 x ) cos ^{2} ( x )}{cos ^{2} ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 2 cos^{4} (x) + 6 sin^{2} (x) - cos (2 x) cos^{2} (x) = 0

Rewrite using trig identities

6 - 4 cos^{4} (x) - 5 cos^{2} (x) = 0

Solve by substitution

cos (x) = 2 i, cos (x) = - 2 i, cos (x) = \frac{3}{2}, cos (x) = - \frac{3}{2}

cos (x) = 2 i :

No Solution

cos (x) = - 2 i :

No Solution

cos (x) = \frac{3}{2} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

cos (x) = - \frac{3}{2} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

2 cos^{2} (x) - 3 sin (x) - 3 = 0

6 tan (θ) + 5 = 0

x, y = arcsin (x)

tan (a) = \frac{5}{12}

2 sin (x) - 3 = - csc (x)