which expression is equivalent to 1/sin(2x) - cos(2x)/sin(2x)? tan(x) -tan(x) 1/cos(x) -1/cos(x)

Explanation:

Step1: Combine the fractions

Since the denominators are the same, $\frac{1}{\sin(2x)}-\frac{\cos(2x)}{\sin(2x)}=\frac{1 - \cos(2x)}{\sin(2x)}$.

Step2: Use double - angle formulas

Recall that $\cos(2x)=1 - 2\sin^{2}x$ and $\sin(2x)=2\sin x\cos x$. Then $1-\cos(2x)=2\sin^{2}x$. So $\frac{1 - \cos(2x)}{\sin(2x)}=\frac{2\sin^{2}x}{2\sin x\cos x}$.

Step3: Simplify the fraction

Cancel out the common factor $2\sin x$ in the numerator and denominator. $\frac{2\sin^{2}x}{2\sin x\cos x}=\frac{\sin x}{\cos x}$.

Step4: Recall the tangent function

Since $\frac{\sin x}{\cos x}=\tan x$, the equivalent expression is $\tan x$.

Answer:

$\tan(x)$