Simplify each expression. See Example 4.

2 ta...

Question

Simplify each expression. See Example 4.

2 tan 15°/(1 - tan² 15°)

Accepted Answer

Hello, everyone. We are asked to simplify the following trigonometric expression. We have two tangent of 75 degrees divided by open parentheses, one minus the tangent squared of 75 degrees. Our answer choices are a negative radical three divided by three B radical three divided by three, see negative radical three. The radical three first recall that the double angle formula for tangent is that the tangent of two A equals two multiplied by the tangent of A divided by one minus the tangent squared of A. In looking at our original expression, this looks like it is the tangent double angle formula. So in our example, a equals 75 degrees. So the tangent of two multiplied by 75 degrees, it is the tangent of 150 degrees. Now that we know this, we can work on finding the value. Unfortunately, a unit circle doesn't have tangent directly on it. So we need to use that tangent is the same as sine divided by cosine. So this would be the sine of 150 degrees divided by the cosine of 150 degrees. Referencing a unit circle. I find that the sign of 150 degrees is one half where the cosine of 150 degrees is negative radical three divided by two. So putting these together sine divided by cosine, I have one half divided by negative radical three divided by two. I recall that in a complex fraction, if both of my fractions have the same denominator, I can cancel them out, they reduce. So this can be rewritten as one divided by negative radical three. I'm going to pull that negative sign out front and deal with the one divided by radical three part. I need to rationalize my denominator. I do not want to leave a radical in the denominator. So I'm going to multiply numerator and denominator by the square root of three. And this will give me negative radical three divided by three. So our value here, our answer is negative radical three divided by three which is answer choice. A have a nice day.

Simplify each expression. See Example 4.

2 tan 15°/(1 - tan² 15°)

Key Concepts

Tangent Function

Pythagorean Identity

Double Angle Formulas

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