Verify that each equation is an identity.
(sec α - tan α...

Question

Verify that each equation is an identity.

(sec α - tan α)² = (1 - sin α)/(1 + sin α)

Accepted Answer

Hello. Today we are going to be be proving whether the given equation is an identity or not. In order to show that this is an identity, we want to prove that the left hand side of the equation will equal to the right hand side of the equation. Now, in order to do this, let's go ahead and first simplify the left hand side of the equation. Notice that the left hand side of the equation is given to us as a squared quantity. If we were to expand the left hand side, we can rewrite the left hand side to be the quantity cos seeking of X minus cotangent of X multiplied by the quantity cos seeking of X minus co tangent of X. If we were to multiply both of these quantities together, we can rewrite the left hand side to be co seek and squared of X minus two multiply by co seeking of X multiplied by cotangent of X plus cotangent squared of X. Now the right hand side of the given equation are in terms of cosign. So what we wanna do is we want to use the trigonometric identities for cosecant and cotangent to rewrite them in terms of sign and cosign the trigonometric identity for coin of X is given to us as one divided by sine of X. And the trigonometric identity for cotangent of X is given to us as cosine of X divided by sine of X. This will give us substitutions for cos seeking and cotangent, but we will still need substitutions for cotangent squared and co seeking squared. Well, if we were to square both sides of the coin identity, we can rewrite the identity to be co seek and squared of X is equal to one divided by sine squared of X. And if we were to do the same with the cotangent identity, we can rewrite the identity to be cotangent squared of X is equal to cosine squared of X divided by sine squared of X. Now we have substitutions for co tangent squared and coin squared and applying these substitutions will allow us to rewrite the left hand side to be one divided by sine squared of X minus two multiplied by one divided by sine of X multiplied by cosine of X divided by sin of X plus cosine squared of X divided by sine squared of X. Now, if we were to multiply the three middle terms together, the three middle terms will simplify to leave us with negative two multiplied by cosine of X divided by sine squared of X. And since all three of the fractions have the same denominator of sine squared of X. We can combine the three fractions together and rewrite the left hand side to be one minus two multiplied by cosine of X plus cosine squared of X all divided by sine squared of X. Now again, the denominator on the right hand side of the equation is in terms of cosine. So what we need to do is we need to use another identity to rewrite sine squared of X to be in terms of cosine. In order to do this, we'll be using the Pythagorean identity sine squared of X plus cosine squared of X is equal to one. If we were to solve for sine squared of X within the identity, we can rewrite the identity to be sine squared of X is equal to one minus cosine squared of X. That means that we can rewrite the denominator to be one minus cosine squared of X. That will allow us to rewrite the left hand side of the equation to be one minus two multiplied by cosine of X plus cosine squared of X divided by one minus cosine squared of X. Now, one thing to note is that the numerator and the denominator are both factor quantities. The numerator will factor to leave us with cosine of X minus one squared. And if we were to expand this quantity, we can rewrite the numerator to be the quantities cosine of X minus one multiplied by cosine of X minus one, the denominator will factor to leave us with one minus cosine of X multiplied by one plus cosine of X. So what can we do from here? Well, if we were to take cosine of X minus one reorder the terms and factor out a negative one, we can rewrite the numerator to be negative one minus cosine of X multiplied by cosine of X minus one, all divided by one minus cosine of X multiplied by one plus cosine of X. We have the quantity one minus cosine of X in both the numerator and denominator. So we can go ahead and cancel those quantities that will leave us on the left hand side, negative cosine of X minus one divided by one plus cosine of X. And if we were to factor or distribute this negative one into the numerator, we can rewrite the numerator to be one minus cosine of X divided by one plus cosine of X. However, keep in mind that the right hand side of the equation is given to us as one plus cosine of X divided by one minus cosine of X. What this means is that the left hand side of the equation does not equal to the right hand side of the equation. Thus proving that this is not an identity. And what this means is that the answer to this problem is going to be B So I hope this video helps you in understanding how to approach this problem and I'll go ahead and see you all in the next video.

Verify that each equation is an identity.
(sec α - tan α)² = (1 - sin α)/(1 + sin α)

Key Concepts

Trigonometric Identities

Secant and Tangent Functions

Algebraic Manipulation

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