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How do you simplify: cos^2x+cos^2(π/2-x) ?

Answer 1

Violet Anderson

Trig identity: #cos (pi/2 - x) = sin x# Therefor, #cos^2 x + cos^2 (pi/2 - x) = cos^2 x + sin^2 x = 1#

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Answer 2

Delilah Aguilar

#1#

#cos^2x+cos^2(π/2-x)#

Can be rewritten as: #cos^2x+cos(π/2-x)*cos(π/2-x)#

Apply cosine difference identity: #cos^2x+(cos(pi/2)*cosx+sin(pi/2)sinx)(cos(pi/2)*cosx+sin(pi/2)sinx)#

Simplify: #cos^2x+(0*cosx+1*sinx)(0*cosx+1*sinx)#

#cos^2x+(sinx)(sinx)#

#cos^2x+sin^2x#

Apply the Pythagorean identity: #1#

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Answer 3

Charles Autterson

[ \cos^2x + \cos^2\left(\frac{\pi}{2} - x\right) ]

Using the trigonometric identity ( \sin^2\theta + \cos^2\theta = 1 ), we can simplify the expression:

[ \cos^2x + \cos^2\left(\frac{\pi}{2} - x\right) = \cos^2x + \sin^2\left(\frac{\pi}{2} - x\right) ]

Recall that ( \sin\left(\frac{\pi}{2} - x\right) = \cos x ), so:

[ \cos^2x + \sin^2\left(\frac{\pi}{2} - x\right) = \cos^2x + \cos^2x ]

Combine like terms:

[ \cos^2x + \cos^2x = 2\cos^2x ]

So, the simplified expression is ( 2\cos^2x ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.