Answer:
Proof provided in the explanation.
Step-by-step explanation:
[tex]\cos^4(\theta)-\sin^4(\theta)+1[/tex]
I'm going to rewrite that middle term so I can apply Pythagorean Identity, [tex]\sin^2(\theta)=1-\cos^2(\theta)[/tex]:
[tex]\cos^4(\theta)-(\sin^2(\theta))^2+1[/tex]
[tex]\cos^4(\theta)-(1-\cos^2(\theta))^2+1[/tex]
Use [tex](a+b)^2=a^2+2ab+b^2[/tex]:
[tex]\cos^4(\theta)-[1-2\cos^2(\theta)+\cos^4(\theta)]+1[/tex]
Distribute:
[tex]\cos^4(\theta)-1+2\cos^2(\theta)-\cos^4(\theta)+1[/tex]
Combine like terms:
[tex]\cos^4(\theta)-\cos^4(\theta)-1+1+2\cos^2(\theta)[/tex]
Applying inverse property of addition:
[tex]0+0+2\cos^2(\theta)[/tex]
Applying identity property of addition:
[tex]2\cos^2(\theta)[/tex]
To prove cos^4x-sin^4x +1= 2 cos^2 x
cos^4x -sin^4 x+1
= ( cos^2x +sin^2x)(cos^2x - sin^2x) +1
You need to remember 3 basic results
sin^2x + cos^2x= 1
cos2x = cos^2x - sin^2x
1+ cos 2x = 2 cos^2x
Now Let's revert back to question
= ( cos^2x + sin^2x )( cos^2x - sin^2x) +1
= 1( cos2x) +1
= cos2x +1
= 2 cos^2x
