SOLUTION: How do I prove the following identity: sin^3x + cos^3x + sinxcos^2x + sin^2xcosx= sinxcosx I believe I need to factor the left side as the first step, especially the first tw

Algebra ->  Trigonometry-basics -> SOLUTION: How do I prove the following identity: sin^3x + cos^3x + sinxcos^2x + sin^2xcosx= sinxcosx I believe I need to factor the left side as the first step, especially the first tw      Log On


   

Question 1205464: How do I prove the following identity:
sin^3x + cos^3x + sinxcos^2x + sin^2xcosx= sinxcosx
I believe I need to factor the left side as the first step, especially the first two terms that involve cube roots, but it just becomes a bit messy for me.
Please let me know if I'm on the right track as per my thoughts on factoring the cubed terms.
Thank you.
Found 2 solutions by greenestamps, math_tutor2020:
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


The equation you show is NOT an identity.

You can see that quickly by trying x=0, where cos(x)=1 and sin(x)=0. The equation then says

0%2B1%2B0%281%29%2B0%281%29=0%281%29
0%2B1%2B0%2B0=0

You apparently have not shown the right side of the equation correctly.



Group the terms in pairs...



...factor out the common factor in each pair...



...and find the common factor (sinx+cosx)...

%28sin%28x%29%5E2%2Bcos%28x%29%5E2%29%28sin%28x%29%2Bcos%28x%29%29

...and then use sin^2x+cos^2x=1:

%281%29%28sin%28x%29%2Bcos%28x%29%29=sin%28x%29%2Bcos%28x%29

The expression on the left is equivalent to sinx+cosx -- not to (sinx)(cosx).

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Tutor greenestamps has a great answer.
To prove or disprove an identity, we can graph the left hand side and right hand sides separately (as separate functions). If one curve overlaps the other completely, then we have an identity.

Desmos graph link
https://www.desmos.com/calculator/nuo8scmnwb
Curve 5 = left hand side of your given equation
Curve 6 = right hand side of your original equation
Curve 7 = what the right hand side should be
Desmos doesn't like the notation sin^3(x) so you'll have to do (sin(x))^3 instead.

Graphs 5 and 6 do not match up.
Therefore this is visual proof that your original equation is not an identity.

Graphs 5 and 7 do match perfectly. Repeatedly click the button for graph 7 so it turns on and off. The curve should blink different colors. It should help show the overlap.
Because of this perfect match, it confirms the identity that greenestamps posted.

---------------------------------------------

Here's another way to visually confirm or disprove an identity.

We use the idea that A = B turns into A-B = 0

If f(x) = g(x), then f(x) - g(x) = 0
I recommend you think of it as f(x) - ( g(x) ) = 0 because g(x) often will contain a plus or minus sign buried somewhere in it.

So you'll subtract the two sides to plot as one single expression. If the curve perfectly overlaps the x axis (i.e. is a flat horizontal line overlapping the x axis), then the two sides are the same and we have an identity.

Here's what it looks like when comparing the two sides of your equation
https://www.desmos.com/calculator/iodyzrhquu
The items subtract to form a curve that doesn't overlap the x axis. Since we don't get a flat horizontal line over the x axis, it proves the original equation is not an identity.