Question

Prove the identity: e^(-sinx cosx tanx) = 1.

a) True
b) False

Answer 1

Upon simplifying the given expression by substituting tanx with sinx/cosx, it becomes clear that e^(-sin^2x) does not equal 1 for all values of x. Therefore, the statement is false.

Step-by-step explanation:

Proving the Identity: e(-sinx cosx tanx) = 1

Let's analyze the expression given: e(-sinx cosx tanx). To prove if the identity is true, we need to simplify the exponent. We know that tanx = sinx/cosx from trigonometric identities. By substituting tanx with sinx/cosx, we have:

e(-sinx cosx (sinx/cosx))

Now, we simplify the expression by canceling out the cosx in the numerator with the cosx in the denominator:

e(-sin2x)

Since the square of sinx can be any real number between 0 and 1, e(-sin2x) cannot equal to 1 for all values of x. Therefore, the statement is false.

The correct answer is that the statement is false and the expression does not always equate to 1.