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Algebra/cosine question

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Question
Hello Scott,
I really need help with this question. I've gone through all of what I know, but I just can't figure it out.

Note: x is used for theta

Write the expression in terms of cos(x). Then, simplify.
cos^4x - sin^4x + sin^2x
Thank you!

Answer
The expression cos^4(x) - sin^4(x) + sin^2(x) can be rewritten as
cos^4(x) - sin^2(x)(sin^2(x) - 1).

Since it is known that sin^2(x) + cos^2(x) = 1,
we can say that -sin^2(x) = cos^2(x) - 1 and sin^2(x) - 1 = -cos^2(x).

Putting these into cos^4(x) - sin^2(x)(sin^2(x) - 1) gives
cos^4(x) + (cos^2(x) - 1)(-cos^2(x)).

Multiplying this out gives cos^4(x) - cos^4(x) + cos^2(x).

The first two terms cancel, leaving only cos^2(x).



Another approach would be to factor the original into
(cos^x(2) - sin^2(x))(cos^2(x) + sin^2(x)) + sin^2(x).

It is known that cos^2(x) + sin^2(x) = 1, so we have
cos^2(x) - sin^2(x) + sin^2(x).

Here, the sin^2(x) terms cancel, leaving cos^2(x).


Either way, there's the answer.

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Scott A Wilson

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I have solved story problems, linear equations, parabolic equations. I have also solved some 3rd order equations and equations with multiple variables.

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