csc(x)/[sec(x) - tan(x)] - csc(x)/[sec(x) + tan(x)] = 2sec(x)
When verifying, or proving, and identity, the common practice is to manipulate one side of an equal sign (usually the more complex side, unless prior known identities dictate otherwise) to become the other side.
When faced with rational expressions, a common strategy is to combine into one rational expression using a common denominator. If the denominators are different, a common denominator can simply be the product of the different denominators. To simplify and streamline the process, if a factor expression appears in the denominators of more than one of the rational expressions to be combined (not the case in our problem), you only need to use it as a factor in your common denominator once, raised to the highest exponent that it appears in any one of the individual rational expression denominators; thus creating the Least Common Denominator (LCD).
In our case, the common denominator will be [sec(x) - tan(x)]*[sec(x) + tan(x)]
To convert any one of the rational expressions to be combined to that of the common denominator, divide the common denominator by the denominator of the rational expression being converted, take the result divided by itself and multiply the rational expression to be converted (anything divided by itself being "1", and anything multiplied by "1" is the same).
So, the above common denominator divided by the denominator of the first term of our problem is
[sec(x) - tan(x)]*(sec(x) + tan(x)]/[sec(x) - tan(x)]=[sec(x) + tan(x)]
AND taking the result divided by itself and multiplying by the first term of our problem, is
{[(sec(x) + tan(x)]/[sec(x) + tan(x)]}*{csc(x)/[sec(x) - tan(x)]}=
{[sec(x) + tan(x)]*csc(x)/{[sec(x) - tan(x)]*[sec(x) + tan(x)]}
AND, the above common denominator divided by the denominator of the second term of our problem is
[sec(x) - tan(x)]*[(sec(x) + tan(x)]/[sec(x) + tan(x)]=[sec(x) - tan(x)]
AND taking the result divided by itself and multiplying by the second term of our problem, is
-{[(sec(x) - tan(x)]/[sec(x) - tan(x)]}*{csc(x)/[sec(x) + tan(x)]}=
-{[sec(x) - tan(x)]*csc(x)/{[sec(x) - tan(x)]*[sec(x) + tan(x)]}
At this point, it is helpful to recognize that the common denominator,
[sec(x) - tan(x)]*[sec(x) + tan(x)]=[sec(x)]^2-[tan(x)]^2 by the difference of two squares.
AND that [sec(x)]^2-[tan(x)]^2 =1 as one version of the Pythagorean Trigonometric identities
(which can be proven by dividing each term of [sin(x)]^2+[cos(x)]^2=1 by [cos(x)]^2)
So, the first term
{[sec(x) + tan(x)]*csc(x)/{[sec(x) - tan(x)]*[sec(x) + tan(x)]}=
{[sec(x) + tan(x)]*csc(x)/1
AND the second term is
{[sec(x) - tan(x)]*csc(x)/{[sec(x) - tan(x)]*[sec(x) + tan(x)]}=
-[sec(x) - tan(x)]*csc(x)/1
AND combining the two terms:
{[sec(x) + tan(x)]*csc(x)/1}-{[sec(x) - tan(x)]*csc(x)/1}=
{[sec(x) + tan(x)]*csc(x)-[sec(x) - tan(x)]*csc(x)}/1=
[sec(x) + tan(x)]*csc(x)-[sec(x) - tan(x)]*csc(x)}=
csc(x){[sec(x) + tan(x)] - [sec(x) - tan(x)]}=
csc(x){sec(x) + tan(x) - sec(x) + tan(x)}
csc(x){2tan(x)}=2[1/sin(x)][sin(x)/cos(x)]=2/cos(x)=2sec(x)
