Question

Please help me simplify. What is `cos^4theta -sin^4theta +sin^2theta` equal to?

Answer 1

`cos^2(x)`

Explanation:

We have
`(cos^4(x)-sin^4(x))=(cos^2(x)-sin^2(x)(cos^2(x)+sin^2(x))+sin^2(x)`
Now we use that
`sin^2(x)+cos^2(x)=1`

So we get
`cos^2(x)-sin^2(x)+sin^2(x)=cos^2(x)`

Answer 2

`cos^2theta`

Explanation:

`cos^4theta-sin^4theta" is a "color(blue)"difference of squares"`

`•color(white)(x)a^2-b^2=(a-b)(a+b)`

`"here "a=cos^2theta" and "b=sin^2theta`

`cos^4theta-sin^4theta=(cos^2theta-sin^2theta)(cos^2theta+sin^2theta)`

`[cos^2theta+sin^2theta=1]`

`cos^4theta-sin^4theta+sin^2theta`

`=cos^2theta-sin^2theta+sin^2theta`

`=cos^2theta`