Verify the identity: (Cos x - sin x)^2 + (cos x + sin x)^2 = 2.

Question:

Verify the identity: {eq}(Cos x - sin x)^2 + (cos x + sin x)^2 = 2. {/eq}

Proving

We have to prove that both sides of the given equation are the same. For this, we need the following identity:

$$\sin^2 x+\cos^2 x=2 $$


We apply this identity after we have expanded the more complicated side of the equation.

Answer and Explanation:


The identity is verified as follows.

$$\begin{align} (\cos x - \sin x)^2 + (\cos x + \sin x)^2& = 2\\ \cos^2 x+\sin ^2 x+2\sin x\cos x +\cos^2 x+\sin ^2 x-2 \sin x \cos x&=2&&&&\left [ \because (a+b)^2=a^2+b^2+2ab ,(a-b)^2=a^2+b^2-2ab\right ]\\ 1+1&=2&&&&\left [ \because \sin^2 x+\cos^2 x=2 \right ]\\ \therefore 2&=2 \end{align} $$


Learn more about this topic:

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Trigonometric Identities: Definition & Uses

from Honors Precalculus Textbook

Chapter 23 / Lesson 1
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