Let's solve the equation step by step:
[tex]\[\frac{\cos \theta}{1-\sin \theta}-\frac{\cos \theta}{1+\sin \theta}= 2 \tan \theta\][/tex]
Step 1: Find a common denominator for the left-hand side expressions.
The denominators are [tex]\(1 - \sin \theta\)[/tex] and [tex]\(1 + \sin \theta\)[/tex].
Common denominator = [tex]\((1 - \sin \theta)(1 + \sin \theta)\)[/tex]
Step 2: Rewrite the left-hand side with the common denominator
[tex]\[
\frac{\cos \theta (1 + \sin \theta)}{(1 - \sin \theta)(1 + \sin \theta)} - \frac{\cos \theta (1 - \sin \theta)}{(1 - \sin \theta)(1 + \sin \theta)}
\][/tex]
Step 3: Simplify the numerators
[tex]\[
\frac{\cos \theta (1 + \sin \theta) - \cos \theta (1 - \sin \theta)}{(1 - \sin \theta)(1 + \sin \theta)}
\][/tex]
[tex]\[
= \frac{\cos \theta (1 + \sin \theta) - \cos \theta (1 - \sin \theta)}{1 - \sin^2 \theta}
\][/tex]
Step 4: Use the Pythagorean identity [tex]\(1 - \sin^2 \theta = \cos^2 \theta\)[/tex]
[tex]\[
= \frac{\cos \theta (1 + \sin \theta) - \cos \theta (1 - \sin \theta)}{\cos^2 \theta}
\][/tex]
[tex]\[
= \frac{\cos \theta + \cos \theta \sin \theta - \cos \theta + \cos \theta \sin \theta}{\cos^2 \theta}
\][/tex]
Step 5: Combine like terms in the numerator
[tex]\[
= \frac{2 \cos \theta \sin \theta}{\cos^2 \theta}
\][/tex]
Step 6: Simplify the fraction
[tex]\[
= 2 \frac{\sin \theta}{\cos \theta}
\][/tex]
[tex]\[
= 2 \tan \theta
\][/tex]
Thus, we have shown that:
[tex]\[\frac{\cos \theta}{1-\sin \theta}-\frac{\cos \theta}{1+\sin \theta}= 2 \tan \theta\][/tex]
The initial equation is verified to be true. So, the given equation holds, and the solution is correct.
