Simplify each expression. See Example 4.
tan 34°/2(1 -...

Question

Simplify each expression. See Example 4.

tan 34°/2(1 - tan² 34°)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to simplify the following trigonometric expression. Our expression is the tangent of 46 degrees divided by two multiplied by the quantity of one minus tangent squared, 46 degrees. Our answer choices are answer choice A one half tangent 92 degrees. Answer choice B 1/4 tangent 92 degrees and answer choice C 1/4 tangent 23 degrees and finally answer trace D the tangent of 92 degrees. All right. So we look at our trigonometric expression and ask ourselves if we see any identities in there. And this looks a lot like the double identity where we have the tangent of two multiplied by an angle. We'll say capital A is equal to two multiplied by the tangent of an angle divided by one minus tangent squared of that angle. So it's very similar. We can see here that our angle A is going to equal 46 degrees. And we've got the tangent of the angle in the numerator and we've got the one minus tangent squared of the angle in the denominator. So what's different? Well, in our given trigonometric expression there's a two being multiplied in the denominator and in our double identity, there's a two being multiplied in the numerator. So how can we make our double identity look like our given trigonometric expression? We're going to do that by multiplying by 1/4. And what we do to one side, we need to do to the other side of the equal sign as well of our double angle identity. So here, if we simplify this, on the left hand side, we can keep this as 1/4 multiplied by the tangent of two A equals. And on the right, the two and the four are going to reduce each other, the two becomes a one and the four becomes a two. So in the numerator, we just have the tangent of our angle A divided by and now it's two multiplied by the quantity of one minus tangent squared of our angle. And so here we can go back substitute in that our angle A is 46 degrees in the three places that it appears here. So on the left, we have 1/4 multiplied by the tangent of two multiplied not by A. Now it's by 46 degrees equals on the right, the tangent of our angle 46 degrees divided by two multiplied by the quantity of one minus the tangent squared of 46 degrees. So now this matches exactly what we were given in the first place. On the right hand side And that means that on the left, we just simplify that. And that's our trigonometric expression simplified. So we can just keep this as a 1/4 multiplied by the quantity of the tangent of and then multiply two by 46 degrees, that's 92 degrees. And that matches the format that our answer choices are using. So 1/4 multiplied by the tangent of 92 degrees that matches with answer choice B well done. We'll catch you on the next one.

Simplify each expression. See Example 4.
tan 34°/2(1 - tan² 34°)

Key Concepts

Tangent Double-Angle Identity

Algebraic Simplification of Trigonometric Expressions

Evaluating Trigonometric Functions at Specific Angles

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