In Exercises 23–26, find the exact value of each expression. Do n...

Question

In Exercises 23–26, find the exact value of each expression. Do not use a calculator.

cos² 𝜋 - tan² 𝜋
        4          4

Accepted Answer

Hello everybody. I hope you are doing all right. Today, today, we're going to be looking at this question that states without using a calculator determine the exact value of the given trigonometric expression where our expression is open bracket, tangent of open parentheses, pi divided by three closed parentheses, closed bracket squared minus open bracket, cosine of open parentheses, pi divided by three closed parentheses, closed bracket squared. Now the answers provided are a four B negative one C 11 divided by four and D one divided by six. Now to tackle this question, I'm going to be doing a simple recall first using the unit circle. So using our unit circle values, we may be able to recall or if you have a unit circle written down, we know that tangent of pi divided by three that is equal to the square roots of three. And if we have cosine of pi divided by three, that is equal to one half, and we obtain these values from the unit circle, which we should be able to recall. Now using that information, we can go ahead and rewrite our equation. So to rewrite, we had first, we had the tangent of pi divided by three. We already know the answer to that. So we'll have open parentheses squared of three closed parentheses squared. Now minus open parentheses. Well, we had the cosine of pi divided by three, which is one half closed parentheses and we have that squared. Now, the squared of three squared is just three, the square root cancels out and we'll have that minus one half squared. When you have a fraction squared, you square the numerator and the denominator. So we'll have three minus open parentheses, one divided by four, close parentheses. Now to turn the three into a fraction to be able to subtract the 1/4 we're going to find a common denominator. Now three as a fraction divided by four, we have 12 divided by four because 12 divided by four is three. So we have 12 divided by four minus one, divided by four, which will leave us with 11 divided by four. And that will be the answer to this question. And we can see how important it is to know our unit circle and know how it works. So we can apply its values to questions like this one. So I hope that was helpful. And until next time.

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In Exercises 23–26, find the exact value of each expression. Do not use a calculator. cos² 𝜋 - tan² 𝜋 4 4