Write each expression in terms of sine and cosine, and then si...

Question

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

(sec θ - 1) (sec θ + 1)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to write the following trigonometric expression in terms of sine and cosine express your answer in terms of X. Only our trigonometric expression is the quantity of the Kent of X minus one multiplied by the quantity of the sine of X plus one. Our answer choices are answer choice. A cosine squared X answer choice B cosine squared X divided by sine X answer choice C cosine X divided by sine X and answer choice D cosine squared X divided by sine squared X. All right. So we're pretty close here to being in terms of sine and cosine. But we've got this cos can X and we recall from previous lessons that the cos can of X is the reciprocal function for the sine function. So that's the same as one divided by the sine of X. So we can replace that in that set of parentheses here instead of Kant X, that's one divided by the sign of X. And then we can bring the rest down. So one divided by sin of X minus one multiplied by the quantity of the sine of X plus one. But we're to express our answer in terms of X only. So we can't have the minus one and the plus one. So what we can do next here is we can foil this, taking our firsts one divided by the sin of X multiplied by the sin of X. That is one outsides one divided by the sine of X multiplied by one is going to be plus one divided by sine of X, taking our insides negative one multiplied by the sine of X is minus the sine of X. And then our last negative one multiplied by positive one. That is going to be minus one and combining like terms here, one minus one. Well, that cancels out that's zero. So now we just have one divided by the sine of X minus the sine of X. Let's combine these because one's a fraction, we need to have common denominator in order to subtract. So looking at the sine of X, we're going to multiply that by sin of X divided by the sine of X in order to get that common denominator of the sine of X. So we'll bring down one divided by sine X minus and then sine X multiplied by sine X is going to be sine squared X in the numerator divided by the denominator, sine X. Now we can combine our numerators with the common denominator. This will be one minus sine squared X all divided by the sine of X, but that one minus sine squared X. That is familiar, we recall previous lessons on Pythagorean identities. And we recall that one minus sine squared X is equal to cosine squared X. So we can have that in our numerator. So this whole thing is equal to in the numerator, cosine squared X divided by the denominator sine X. And that's it. We look at our answer choices and this matches with answer choice. B well done. We'll catch you on the next one.

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
(sec θ - 1) (sec θ + 1)

Key Concepts

Definition of Secant in Terms of Cosine

Algebraic Simplification of Trigonometric Expressions

Pythagorean Identity

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