Verify that each equation is an identity.
(1 + sin x + c...

Question

Verify that each equation is an identity.

(1 + sin x + cos x)² = 2(1 + sin x) (1 + cos x)

Accepted Answer

Hello, everyone. We are asked if the given equation is an identity we are given in parentheses. One plus the C can of X minus the tangent of X close parentheses squared equals two C can't squared X plus two multiplied by the parentheses. C can't X plus C can't X tangent X plus tangent X closed parentheses. Our answer choices are A yes, B no. For this equation to be an identity, we need to be able to make the left hand side look like the right hand side or vice versa. I'm gonna start with the left hand side. On the left hand side, we have one plus the C cant of X minus the tangent of X closed parentheses squared. First, I wanna do the square on this and I know there is an identity but I honestly almost never remember it. So I'm gonna multiply it out long because I think that's what most people would end up doing here. So this is like saying one plus the C of X minus the tangent of X multiplied by one plus the C of X minus the tangent of X. So multiplying this out first terms one multiplied by one is one, one multiplied by the C can of X is plus C can't X one multiplied by the negative tangent of X is minus 10. X. Next C can't multiplied by one is plus the C can't of X C can of X multiplied by C can of X will give us plus C can't squared of X C can of X multiplied by negative tangent of X will give us negative C can of X tangent of X. And then negative tangent of X multiplied by one will be minus the tangent of X negative tangent of X multiplied by C can of X be negative C of X tangent of X negative tangent of X multiplied by the negative tangent of X will give us a positive tangent squared X. Now I'm gonna combine my like terms. I only have 11 that start with, I know that I wanna pull all of my squares first. So I'm going to take the C can't squared of X and rate that next. So I have one plus the C can't squared of X plus the tangent squared of X. And then looking for my C cans I have a positive C can of X and another positive C can of X. So that's plus two. See of X, I have a negative tangent of X and another negative tangent of X. So minus two tangent of X. And what's left, I have a negative C can't of X tangent of X and another negative C can of X tangent of X. So minus two C can't of X tangent of X. All right. So we're on the right track. Next, I wanna see what else can be simplified. I recall that we have an identity that states that the C can't squared of X is equal to one plus the tangent squared of X. So I'm gonna use that because if I rearrange the first couple terms a little bit, I have both of those things that I need. So I'll have C can't squared of X plus C can't squared of X plus two C can't X minus two tangent of X minus two C can't of X tangent of X. So combining my like terms, I have two C can't squared of X and then our right hand side has a two factored out of the rest of the term. So let's factor that two out. So it'll be plus two open parentheses. C can of X minus C can of X tangent of X minus tangent of X. And that's about all I can do. In this case, in the parentheses, we have different signs for the C of X tangent of X and tangent of X. So for this example, it's answer choice B no, it is not an identity. Have a nice day.

Verify that each equation is an identity.
(1 + sin x + cos x)² = 2(1 + sin x) (1 + cos x)

Key Concepts

Trigonometric Identities

Algebraic Expansion

Factoring

Watch next