to identify the trig identity working from left to right: so you have to work out the left hand side first to see if the left hand side is equal to the right hand side

Question

_{6 tan (x) / cos(x) = 3/ 1- sin(x) - 3/1+sin(x)}

Answer 1

The easy way for this trig identity is to work the right side to make even to the left side. Simple check will do:

3/(1 - sin(x)) - 3/(1 + sin(x))

= [3(1 + sin(x)) - 3(1 - sin(x))] / [(1 + sin(x)(1- sin(x)]

...

= 6tan(x)/cos(x).

The opposite, to make left side equal to right side is not that obvious.

Start with the left side, and we express tan(x) as sin(x)/cos(x).

6tan(x)/cos(x)

= 6[sin(x)/cos(x)]/cos(x)

then we use the Pythagorean trig identity to express cos²(x) = 1 - sin²(x)

= 6sin(x)/[1 - sin²(x)]

and we factor out the denominator:

= 6sin(x)/[(1 + sin(x)(1-sin(x)]

then we modify the numerator by adding and subtracting 3 and breaking 6sin(x) into sum of 3sin(x) and 3sin(x):

6sin(x) = 3 + 3 sin(x) - 3 + 3sin(x)

=3(1+sin(x)) - 3(1 - sin(x))

You then can finish the factoring and will get the identity proven that way.

Note that you do not need to use any particular side. You always can start with the side looking easy and go to the other side. Once you finish, you can try working back way to achieve the desired result.

Some of the steps have been omitted for you to practice. Any questions?