In Example 6, change the first term on the left to sin y/cos³ y .

Data from Example 6

Prove the identity

Because the more complicated side is on the left, we will change the left side to tan y, the form on the right. Because we want tan y as the final form, we will use the basic identities to introduce tan y where possible. Thus, noting the cot y in the denominator, we substitute 1/tan y for cot y. This gives

In the form on the right we now note the factor of tan y in each term. Factoring tan y from each of these terms, and then using the basic identity 1 + tan² y = sec² y in the form sec² y − tan² y = 1, we can complete the proof. Therefore,

sec y cot y tan y = tany.