cos(2x)/1−sin(2x) = cos x+sin x/cos x−sin x
To start, let's simplify the left side of the equation.
We know that cos(2x) = cos^2(x) - sin^2(x), so we can rewrite the left side as:
(cos^2(x) - sin^2(x)) / (1 - sin(2x))
Now, let's simplify the right side of the equation.
We can rewrite the right side as (cos(x) + sin(x)) / (cos(x) - sin(x)).
Now, let's solve for the common denominator.
Multiplying both the numerator and denominator of the left side by (1 + sin(2x)) and multiplying both the numerator and denominator of the right side by (cos(x) + sin(x)), we have:
(cos^2(x) - sin^2(x)) * (1 + sin(2x)) / [(1 - sin(2x)) * (1 + sin(2x))] = (cos(x) + sin(x)) * (cos(x) + sin(x)) / [(cos(x) - sin(x)) * (cos(x) + sin(x))].
Simplifying further:
cos^2(x) - sin^2(x) * (1 + sin(2x)) = (cos(x) + sin(x))^2.
Expanding the left side:
cos^2(x) - sin^2(x) - sin^2(x) * sin(2x) = (cos(x) + sin(x))^2.
Now, let's expand the right side:
cos^2(x) + 2 * cos(x) * sin(x) + sin^2(x) = cos^2(x) + 2 * cos(x) * sin(x) + sin^2(x).
Since the left side is equal to the right side, the equation is true for all values of x.