Question

Verify the identity: 1 + sec(-x)/sin(-x) + tan(-x) = -csc(x).

A) True
B) False

Answer 1

The identity 1 + sec(-x)/sin(-x) + tan(-x) = -csc(x) is False after rewriting the trigonometric functions of negative angles in terms of positive angles and simplifying.

Step-by-step explanation:

To verify the identity 1 + sec(-x)/sin(-x) + tan(-x) = -csc(x), let's rewrite the trigonometric functions of negative angles in terms of positive angles.

Recall the following properties for trigonometric functions of negative angles:

Substituting these into the original identity, we get:

1 + (1/cos(x))/(-sin(x)) - tan(x)
= 1 - (1/cos(x))/sin(x) - sin(x)/cos(x)
= 1 - sec(x) * csc(x) - csc(x)
Since sec(x) * csc(x) is 1/cos(x) * 1/sin(x), which simplifies to 1/(sin(x) * cos(x)), and since 1 - sin(x) * cos(x) is not equivalent to - csc(x), the original statement is False.