Question

Factor the trigonometric expression and simplify. tan4x - sec 4x

A. sec²x
B.-2 tan²x - 1
C. tan 2 - sec²x
D. sec²x + tan²x

Answer 1

The trigonometric expression tan4x - sec4x simplifies to -2tan2x - 1 after recognizing the difference of squares and using a trigonometric identity for sec2x.

Step-by-step explanation:

To factor the trigonometric expression tan4x - sec4x, we start by recognizing a difference of squares. The expression can be written as:

(tan2x + sec2x)(tan2x - sec2x)

However, we know that sec2x = 1 + tan2x, which we can substitute into our expression.

So by substituting, our expression becomes:

(tan2x + (1 + tan2x))(tan2x - (1 + tan2x))

Simplified further, it results in:

(2tan2x + 1)(-1)

Finally, the expression simplifies to -2tan2x - 1, which is represented by option B.