Simplify the lefthandside so that $L H S=R H S$ :
\[
\frac{\cos (t)}{1+\sin (t)}-\frac{\cos (t)}{1-\sin (t)}=-2 \tan (t)
\]

Step 1 ：Simplify the lefthandside so that \(L H S=R H S\) : \(\frac{\cos (t)}{1+\sin (t)}-\frac{\cos (t)}{1-\sin (t)}=-2 \tan (t)\)

Step 2 ：The first step to simplify the left hand side of the equation is to find a common denominator for the two fractions. The common denominator would be the product of the two denominators, which is \((1+\sin(t))(1-\sin(t))\).

Step 3 ：After finding the common denominator, we can combine the two fractions into one. Then, we can simplify the numerator and denominator separately.

Step 4 ：After simplifying, we can check if the left hand side equals to the right hand side.

Step 5 ：After simplifying the left hand side of the equation, we find that it equals to \(-2 \tan (t)\), which is the same as the right hand side of the equation. Therefore, the equation holds true.

Step 6 ：Final Answer: The simplified form of the left hand side of the equation is \(\boxed{-2 \tan (t)}\), which equals to the right hand side of the equation.