since cos^2(x) = 1 - sin^2(x), your equation becomes:
2 * (1 - sin^2(x)) = sin^2(x)
simplify to get:
2 - 2*sin^2(x) = sin^2(x)
add 2*sin^2(x) to both sides of the equation to get:
2 = sin^2(x) + 2*sin^2(x) which results in:
2 = 3*sin^2(x)
divide both sides of this equation by 3 to get:
2/3 = sin^2(x)
take the square root of both sides of this equation to get:
sin(x) = plus or minus sqrt(2/3)
take the arcsin of sqrt(2/3) to get:
x = 54.73561032 degrees.
take the arcsin of - sqrt(2/3) to get:
x = - -54.73561032 degrees.
the sine is positive in quadrant 1 and 2.
the angle in quadrant 1 is 54.73561032 degrees.
the angle in quadrant 2 is 180 - 54.73561032 which is equal to 125.2643897 degrees.
the sine is negative in quadrant 3 and 4.
the angle in quadrant 3 is 180 + 54.73561032 which is equal to 234.7356103 degrees.
the angle in quadrant 4 is 360 - 54.73562032 which is equal to 305.2643897.
note that an angle of 305.2643897 degrees is the same angle as - 54.73561032 degrees, i.e. they are the identical position on the standard trig function graph.
confirm your answer is correct by substituting in the original equation for x.
your original equation is:
2cos^2(x) = sin^2(x)
when x = 54.73561032 degrees:
sin^2(x) = 2/3
2cos^2(x) = 2/3
they are equal.
when x = 125.2643897 degrees:
sin^2(x) = 2/3
2cos^2(x) = 2/3
they are equal.
when x = 234.7356103 degrees:
sin^2(x) = 2/3
2cos^2(x) = 2/3
they are equal.
when x = 305.2643897 degrees:
sin^2(x) = 2/3
2cos^2(x) = 2/3
they are equal.
a graph of both equations is shown below:
you can see on the graph that the equation of sin^2(x) intersects with the equation of 2cos^2(x) at the indicated spots on the graph which is equal to y = 2/3 in all cases.
