Final answer:
The trigonometric identity (tanx - secx)(tanx + secx) is verified by expanding it using difference of squares and simplifying to show that it is equal to -1.
Step-by-step explanation:
The student is asking for a verification of the trigonometric identity involving tanx and secx. To verify the identity, we must show that (tanx - secx)(tanx + secx) simplifies to a known identity. We can use the difference of squares, a strategy from algebra, to expand this expression:
(tanx - secx)(tanx + secx) = tan2x - sec2x
We know from trigonometry that sec2x is equal to tan2x + 1. So, when we replace sec2x in the expanded form, we get:
tan2x - (tan2x + 1)
By simplifying, we have:
tan2x - tan2x - 1 = -1
Thus, the identity (tanx - secx)(tanx + secx) = -1 is verified, confirming the student's identity as true.