VERIFYING TRIGONOMETRIC IDENTITIES
You can use the basic trigonometric identities, along with the definitions of the trigonometric functions, to verify other identities. for example, suppose you wish to know if sin o sec o cot o = 1 is an identity. To find out, simplify the expression on the left side of the equation by using the identities and definitions.
sin o sec o cot o = 1
sin o . 1/cos o . 1/tan o = 1 (sec o = 1/cos 0 and cot o = 1/tan o)
sin o / cos o . 1/tan o = 1 (multiply sin o and 1/cos o)
tan o . 1/tan o = 1 (tan o = sin o/cos o)
1 = 1 (tan o/tan o = 1)
Thus, sin o sec o cot o = 1 is an identity.
in a way, verifying an identity is like checking the solution to an equation. You do not know if the expression on each side are equal. That is waht you are trying to verify. so, you must simplify one or both sides of the sentence separately until they are the same. (Often it is easier to work with only one side of the sentence. you may choose either side.)
The following suggestions are helpful in verifying trigonometric identities. Study the example to see how these suggestions can be used to verify an identity.
Start with the more complicated side of the equation. Transform the expression into the form of the simpler side.
Work with each side of the equation at the same time. Transform each expression separately into the same form.
Substitute one or more basic trigonometric identities to simplify the expression.
Multiply both numerator and denominator by the same trigonometric expression.
There is often more than one way to verify an identity. Remember that verifying an identity is not the same as solving an equation. An identity is true for all values of the variable except those values for which either side of the equation is undefined.