Solve for 0 less than or equal to x less than or equal to 360.
a) cos2x = tan2x
b) cos3x + cosx = 0
Many thanks in advance.
a) $\displaystyle \cos{2x} = \tan{2x}$
$\displaystyle \cos{2x} = \frac{\sin{2x}}{\cos{2x}}$
Multiply both sides by $\displaystyle \cos{2x}$
$\displaystyle \cos^2{2x} = \sin{2x}$
Rearrange and use the Pythagorean identity sin^2(x) + cos^2(x) = 1:
$\displaystyle 0 = \sin^2{2x} + \sin{2x} - 1$
Use the quadratic formula:
$\displaystyle \sin{2x} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
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Although it's kind of long, but you have to use the identities:
$\displaystyle \cos{(A\pm B)} = \cos{A}\cos{B} \mp \sin{A}\sin{B}$
$\displaystyle \sin{2x} = 2\sin{x}\cos{x}$
Keep expanding the cosines/sines until you get everything in the same angle.