Verify that each equation is an identity.
sin² x(1 + cot...

Question

Verify that each equation is an identity.

sin² x(1 + cot x) + cos² x(1 - tan x) + cot² x = csc² x

Accepted Answer

Hey, everyone here we are asked is the given equation and identity, we are given sine of X multiplied by the quantity of one minus cotangent of X plus cosine of X multiplied by the quantity of one plus tangent of X minus cos squared of X is equal to two multiplied by cosine of X minus cos squared of X. We have two answer, choice options, answer a yes or answer B no. So in order to determine if our equation is an identity, we need to verify that the left hand side is equivalent to the right hand side. And so taking a look at our left hand side of the equation again, recalling we have sign of X multiplied by the quantity of one minus cotangent of X plus cosine of X multiplied by the quantity of one plus tangent of X minus Kent squared of X. And so, in order to determine if the left hand side is equivalent to the right hand side, we need to begin by reducing this side or the left hand side as much as possible. So we can do this by first rewriting cotangent of X and tangent of X So we first need to recall that cotangent of X is equivalent to cosine of X divided by sin of X. Then we also need to recall that tangent of X is equivalent to sine of X divided by cosine of X. And now we just want to substitute in these equivalent expressions into the left hand side expression. And so we have sine of X multiplied by the quantity of one minus cosine of X divided by sin of X plus cosine of X multiplied by one plus sine of X divided by cosine of X. Then we have minus cos squared of X. And so now we want to apply the distributive property so we can distribute in sin of X and cosine of X to continue to simplify our expression. And so we have sine of X minus cosine of X plus cosine of X plus sine of X minus cos squared of X. And now we just want to combine like terms and we can immediately notice that the cosines will cancel each other out. And so we are just left with two multiplied by sine of X minus coin squared of X. Now, with the left hand side of the equation, fully simplified, we can compare it to the right hand side. And we notice that instead of having two multiplied by cosine of X, we instead have two multiplied by sine of X. So we can clearly see that the left hand side is not equivalent to the right hand side. Therefore, we can conclude that our given equation is not an identity. Therefore, we are left with answer choice B where our answer is no, our given equation is not an identity. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Verify that each equation is an identity.
sin² x(1 + cot x) + cos² x(1 - tan x) + cot² x = csc² x

Key Concepts

Trigonometric Identities

Reciprocal Functions

Simplification Techniques

Watch next