Prove the following identities.

Question

$\frac{\sin heta}{1+\cos heta}=\frac{1-\cos heta}{\sin heta}$

Accepted Answer

Hey everyone. In this problem we're asked if the trigonometric equation, cos a divided by 1+in a equals 1 sin a all divided by cos a is an identity. We have 2 answer choices, option a, it is an identity, and option b, it is not an identity. So when we're dealing with an equation like this and we want to figure out if it's an identity, what it's really asking is whether or not this equation holds true for any value of a for which it's defined. Okay, so let's go ahead. We're gonna start by rewriting this equation. So we have cosine of a divided by 1+sinofa. That's gonna be equal to 1 sinofa divided by cosine of a. Now we can see that there are gonna be some points where this equation is not defined. Okay. If the denominator is equal to 0 on either side of the equal sign, then our function or equation is going to be undefined. Okay, but we will now look at all of the other points. Where this equation is defined, is it always true? So let's go ahead. We're gonna multiply both sides by cosine of a. We're gonna multiply both sides by 1+ine of a. And when we do that, what we're gonna have is cosine of a multiplied by cosine of a is equal to 1 minus sine of a multiplied by 1+ine of But we're just gonna keep simplifying. We're gonna simplify on the left hand side, we're gonna simplify on the right hand side, and we're gonna get down and see whether or not this equation is true just for a couple of values for a or whether it's true no matter what we substitute in for a. So on the left we can rewrite this as cosine squared of a. On the right hand side we're going to expand our brackets. We get 1 plus sine of a, minus sine of a, minus sine squared a. Now our sine a minus sine a, that's gonna add to 0. Okay, so we're left with cosine squared of a is equal to 1 minus sine squared a, And recall that if we have sine squared x plus cosine squared x, then this is equal to 1. Okay, that's one of the very important identities that we want to remember in trig. Now if sine squared plus cosine squared is equal to 1, we could subtract sine squared from both sides, and that tells us that cosine squared of x is gonna be 1 minus sine squared of x. So on the right hand side, our one minus sine squared a, we can just write that as cos squared a. So what we have is that cosine squared of a is equal to cosine squared of a. Okay? That's a check mark. That's true for all values of a that the original equation is defined for. So this equation is indeed an identity. It doesn't matter what a value we use, we have the equation satisfied. So the correct answer here is option A. Thanks everyone for watching. I hope this video helped. See you in the next one.

Prove the following identities.
$\frac{\sin\theta}{1+\cos\theta}=\frac{1-\cos\theta}{\sin\theta}$

Key Concepts

Trigonometric Identities

Algebraic Manipulation

Common Denominators

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