12\$ sin^2 heta\$\$ cos^2 heta\$

I am lost on exactly how to execute this. Help would be lot appreciated.

You are watching: Simplify: (sin θ − cos θ)2 + (sin θ + cos θ)2

Hint:

1) broaden : \$(a+b)^2\$ and also \$(a-b)^2\$ because that all actual numbers \$a\$ and also \$b\$.

2) What is the value of \$sin^2(x) + cos^2(x)\$ for all actual number \$x\$ ?

All you have to do is main point it out.

\$(sin heta - cos heta)^2 + (sin heta + cos heta)^2\$

\$= (sin heta - cos heta)(sin heta - cos heta) + (sin heta + cos heta)(sin heta + cos heta)\$

\$= sin^2 heta - 2sin hetacos heta + cos^2 heta + sin^2 heta + 2sin hetacos heta + cos^2 heta\$

\$=sin^2 heta + cos^2 heta + sin^2 heta+ cos^2 heta\$

\$= 1 + 1\$

\$= 2\$

\$\$eginalign&phantom=left(sin x-cos x ight)^2+left(sin x+cos x ight)^2\&=sin^2x-2sin xcos x+cos^2x+sin^2x+2sin xcos x+cos^2x\&=2sin^2x+2cos^2x\&=2left(sin^2x+cos^2x ight)\&=2endalign\$\$

Since this is a multiple-choice question, girlfriend can very quickly narrow under the price by plugging in values of \$ heta\$. For example, when \$ heta = 0\$, the expression i do not care \$(0-1)^2+(0+1)^2 = 2\$.

Now, looking in ~ the four choices and examining at \$ heta = 0\$, us have:

\$1\$\$2\$\$sin^2 heta = 0^2 = 0\$\$cos^2 heta = 1^2 = 1\$

The prize is plainly the second option.

Thanks for contributing an answer to surfacetoairnewyork.com Stack Exchange!

But avoid

Asking because that help, clarification, or responding to other answers.Making statements based on opinion; earlier them increase with references or an individual experience.

Use surfacetoairnewyork.comJax to style equations. surfacetoairnewyork.comJax reference.

See more: Bail And Bond Difference - What Are The Differences Between Bond And Bail