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Prove that:cos^2(45+A)+cos (45-A)=1​

User KarlM
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2 Answers

4 votes

Explanation:

Prove that


\cos^2(45+A)+\cos^2(45-A) =1

We know that


\cos (\alpha \pm \beta) = \cos \alpha\cos \beta \mp \sin \alpha \sin\ beta)

We can then write


\cos (45+A)=\cos 45\cos A - \sin 45\sin A


\:\:\:\:\:\:\:\:= (√(2))/(2)(\cos A - \sin A)

Taking the square of the above expression, we get


\cos^2(45+A) = (1)/(2)(\cos^2A - 2\sin A \cos A + \sin^2A)


= (1)/(2)(1 - 2\sin A\cos A)\:\:\;\:\:\:\:(1)

Similarly, we can write


\cos^2(45-A) =(1)/(2)(1 + 2\sin A\cos A)\:\:\;\:\:\:\:(2)

Combining (1) and (2), we get


\cos^2(45+A)+\cos^2(45-A)


= (1)/(2)(1 - 2\sin A\cos A) + (1)/(2)(1 + 2\sin A\cos A)


= 1

User Graham Streich
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8.5k points
4 votes

Explanation:


\boxed{cos^2x=(1-cos2x)/(2)}\\cos^2(45+A)+cos^2(45-A)=(1-cos2(45+A))/(2)+(1-cos2(45-A)/(2)\\=(1 - cos(90 +2A) )/(2) + (1 - cos(90 - 2A) )/(2) \\ = (2- ( - sin 2A) - sin2A)/(2) \\ = (2 + sin2A -sin2A )/(2) \\ = (2)/(2) \\ = 1

User Ilene
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8.9k points

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