In Exercises 25–30, use an identity to find the value of each ...

Question

In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator. sec² 23° - tan² 23°

Accepted Answer

Hello everybody. I have we doing tonight today today, we're going to be looking at this question that states without using a calculator determine the value of the following trigonometric expression where the expression is negative tangent, negative tangent squared, 57.1 degrees plus C squared of 57.1 degrees. And the answer choices provided are a zero B one C negative two and D one half. Now, the way to solve this without using a calculator is through different identities. No, let's recall that tangent of theta is equal to sine of theta divided by cosine of theta and see the is equal to one divided by cosine theta. So with that information, we can already rewrite our expression and we have negative open parentheses sine of 57.1 degrees divided by cosine of 57.1 degrees closed parentheses, squared plus open parentheses, one divided by cosine of 57.1 degrees closed parentheses squared. And because we have the same denominator on both of these fractions, we can combine them into one fraction. So this will be equal to open bracket or we have a negative outside. And then open bracket on the numerator will have open parentheses, sign of 57.1 degrees closed parentheses squared plus one, divided by open parentheses, cosine of 57.1 degrees, close parentheses squared. And then we close the bracket. Now we can do another recall and this will be of another identity. So we have to recall that sine squared, theta plus cosine squared data is equal to one. And you might recognize this as a Pythagorean identity. So if we arrange this identity, we, this means that cosine squared data is equal to one minus sine squared theta. So with that information, we can go ahead and rewrite once again. And in the numerator of our fraction where we have a negative outside and we have a sign uh squared of theta plus one. I'm going to rearrange that to now in the numerator instead of I'm going to distribute the negative to the sign and I'll have one minus sign of 57.1 degrees closed parentheses squared. And all that will be the numerator and that'll be divided by open parentheses, cosine of 57.1 degrees closed parentheses squared. And now if we look at the numerator, we see that we have the identity, we just talked about one minus sine square data. So that will be equal to cosine squared data, which means that the numerator now will have cosine of 57.1 degrees squared. So open parentheses, cosine of 57.1 degrees closed parentheses, squared, divided by open parentheses, cosine of a 57.1 degrees closed parentheses squared. And we have the same numerator as we do in the denominator, they're the same number. So when you have the, the same number and the numerator and the denominator, this means that our fraction is equal to one. So after all of that, using different identities and rearranging our problem, we've reached an answer one which is answer choice B so I hope that was helpful. And until next time.

In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator. sec² 23° - tan² 23°

Key Concepts

Pythagorean Identity for Secant and Tangent

Definition of Secant and Tangent Functions

Using Identities to Simplify Expressions

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