Hey, everyone. In this problem, we're asked to verify the identity using our sum and difference identities. And the equation that we're given here is the cotangent of 90 degrees minus theta is equal to the tangent of theta. Now remember, when verifying an identity, we want to start with our more complicated side. And here, that's my left side because I see that I have a difference in that argument. So let's go ahead and get started with simplifying that left side.
Now on this left side, remember that we want to be constantly scanning for identities using our strategies over there. And the cotangent using our reciprocal identity is 1 over the tangent. So here, I can rewrite this as 1 over the tangent of 90 degrees minus theta. Now here your first instinct might be to go ahead and use your difference formula for the tangent, but let's think about what would happen if we did that. Here, since in our argument we have 90 degrees minus theta, that tells me that I'm going to end up taking the tangent of 90 degrees somewhere in that formula. But the tangent of 90 degrees is an undefined value. And remember that whenever we're using our sum and difference formulas for the tangent, if we end up with an undefined value, we actually want to go ahead and rewrite everything in terms of sine and cosine and go from there. That way, we don't end up with an undefined value.
So here, instead of rewriting this cotangent as 1 over the tangent and then using our tangent formula, we're going to instead rewrite the cotangent in terms of sine and cosine. So let's go ahead and do that here. We're going to start from scratch, not with our tangent, and we're going to take this cotangent and break it down in terms of sine and cosine so that we don't end up with an undefined value.
Now remember that the cotangent is equal to the cosine over the sine. So here I'm going to be taking the cosine of 90 degrees minus theta over the sine of 90 degrees minus theta. Now from here, I can use my sum and difference formulas for sine and cosine. So in my numerator, since I have the cosine of 90 degrees minus theta, I'm going to use my difference formula for cosine and since these are being subtracted in the argument I'm going to be adding my two terms together in that formula.
cos ( θ ) = cos ( 90° - θ ) × cos ( θ ) + sin ( 90° ) × sin ( θ )So expanding this out gives me the cosine of 90 degrees times the cosine of theta plus the sine of 90 degrees times the sine of theta. Then in my denominator, I have the sine of 90 degrees minus theta. So I'm going to use my difference formula for sine.
sin ( θ ) = sin ( 90° - θ ) × cos ( θ ) - cos ( 90° ) × sin ( θ )Now here, since we're subtracting in that argument, we're going to be subtracting our two terms in that identity. Now we have a bunch of values that we know from our knowledge of the unit circle. The cosine of 90 degrees is 0, that tells me that these entire terms will go away because they are both 0. And the sine of 90 degrees is simply equal to 1. So I'm just left with the sine of theta in that numerator, which is this term right here, over the cosine of theta, which is that term that's left in the denominator.
Now what is the sine of theta over the cosine of theta? Well, it's simply equal to the tangent of theta. Now looking back at that right side of my equation, remembering that my goal is for both of these sides to be equal, I see that that right side is already the tangent of theta. So pulling that all the way down here, I have successfully verified this identity, showing that the tangent of theta is indeed equal to the tangent of theta having expanded this out using my sum and difference formulas. So now we have successfully verified this identity, which you might recognize as being one of our cofunctional identities that we've used earlier in this course. Let me know if you have any questions, and thanks for watching.