Math  /  Trigonometry

QuestionVerify that the equation given below is an identity. (Hint: cos2x=cos(x+x)\boldsymbol{\operatorname { c o s }} 2 x=\cos (x+x).) cos2x=cos2xsin2x\cos 2 x=\cos ^{2} x-\sin ^{2} x
Rewrite the expression on the left to put it in a more useful form cos2x=cos2x=cos2x=cos(xx)sin(π2x)cos(x+x)1sin2xcos2x\begin{array}{l} \cos 2 \mathrm{x}= \\ \cos 2 \mathrm{x}= \\ \cos 2 x= \\ \cos (x-x) \\ \sin (\pi-2 x) \\ \boldsymbol{\operatorname { c o s }}(\mathrm{x}+\mathrm{x}) \\ 1-\sin 2 x \\ -\cos 2 x \end{array}

Studdy Solution

STEP 1

1. We are given the trigonometric identity cos2x=cos2xsin2x\cos 2x = \cos^2 x - \sin^2 x.
2. We need to verify this identity using the hint provided.

STEP 2

1. Use the hint to express cos2x\cos 2x in terms of cos(x+x)\cos(x+x).
2. Apply the cosine addition formula.
3. Simplify the expression to match the right-hand side of the identity.

STEP 3

Use the hint to express cos2x\cos 2x as cos(x+x)\cos(x+x).
cos2x=cos(x+x) \cos 2x = \cos(x+x)

STEP 4

Apply the cosine addition formula:
cos(a+b)=cosacosbsinasinb \cos(a+b) = \cos a \cos b - \sin a \sin b
Substitute a=xa = x and b=xb = x:
cos(x+x)=cosxcosxsinxsinx \cos(x+x) = \cos x \cos x - \sin x \sin x

STEP 5

Simplify the expression:
cos(x+x)=cos2xsin2x \cos(x+x) = \cos^2 x - \sin^2 x
This matches the right-hand side of the given identity:
cos2x=cos2xsin2x \cos 2x = \cos^2 x - \sin^2 x
The identity cos2x=cos2xsin2x\cos 2x = \cos^2 x - \sin^2 x is verified.

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