Trigonometry

Question

Solve the equation on the interval [0,360]

cos²2x=cos2x+sin²2x

Answer 1

1.  replace (sin 2x)² by(1 - cos²2x)

2.  let u = cos 2x

3. rewrite the equation as u² = u + 1 - u²

4. rearrange & factor the quadratic, giving u = -½  or u = 1

5. go back to definition of u, and write cos 2x = -½ or cos 2x = 1

6. the solutions are 2x = 2pi/3 + k(2pi) or 2x = 4pi/3 + k(2pi)   OR  2x = k(2pi)

7. Therefore x = pi/3 + k(pi)  or x = 2pi/3 + k(pi)  OR x = k(pi).