Verify that each equation is an identity.
(sec α + csc α...

Question

Verify that each equation is an identity.

(sec α + csc α) (cos α - sin α) = cot α - tan α

Accepted Answer

Hello, everyone. We are asked if the given equation is an identity. We are given open parentheses. Sine X plus CC. Can't X closed parentheses, multiplied by the parentheses, cosine X minus C can't X closed parentheses equals sine X cosine X minus tangent X plus cotangent X plus C can't X cosine X. Our answer choices are a yes, B no. For this to be an identity, we need to make the left hand side match the right hand side or vice versa. Here, I'm gonna start with the left hand side. So we have in parentheses, the sine of X plus the Kent of X multiplied by the cosine of X minus C of X. I have two binomials that are being multiplied together. So I'm going to foil them. So first, I'll have the sine of X multiplied by the cosine of X outer will be sin of X multiplied by the negative C can of X which will give me negative sign XC can't X inner, I'll have the coi can of X multiplied by the cosine of X. So that's a positive cosine XKC can X and last, I will have the KC can of X multiplied by the negative C cant of X which is negative C can't XK can X. So from here, I'm going to change my C cans and cos cans into terms of sign and cosine. So my first term Sinex co Sinex does not change. In my second term. I'll have the sine of X multiplied by one divided by the cosine of X. In my third term, I will now have the cosine of X multiplied by one divided by the sine of X. And in my last term, I will have one divided by the cosine of X multiplied by one divided by the sine of X. So let's see if, when I multiply these things together, anything helps me out. So the sine of X cosine of X stays put the second term I'll have minus and it'll be the sine of X divided by the cosine of X. Third term will add the cosine of X divided by the sine of X. And the last term we just end up with one divided by the cosine of X sin of X. I recall that sine of X divided by cosine of X will give me the tangent of X. So I can rewrite this as the sin of X cosine of X minus the tangent of X. And then the cosine of X divided by the sine of X is the same as the cotangent of X. And that last term, I think I'm just gonna put it back to what it began with as the C can of X cosecant of X. So looking at what I have, does it match what's on the right hand side? Almost all of it does till that very last term. But Koi can of X and cosine of X are not the same and I have no way of making them the same. So in this case, this is answer choice B no, it is not an identity. Have a nice day.

Verify that each equation is an identity.
(sec α + csc α) (cos α - sin α) = cot α - tan α

Key Concepts

Trigonometric Identities

Reciprocal Functions

Algebraic Manipulation

Watch next