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Chapter 5: Problem 72

Verify the identity. $$\cos \left(\frac{\pi}{3}+x\right)+\cos \left(\frac{\pi}{3}-x\right)=\cos x$$

Short Answer

Expert verified
The left-hand side of the equation simplifies to \( \cos(x) \) which matches the right-hand side, thus verifying the identity.

Step by step solution

01

Apply the Trigonometric Addition and Subtraction Formulas

Start by applying the formula \( \cos(A+B) = \cos(A)\cos(B) - \sin(A)\sin(B) \) to \( \cos\left( \frac{\pi}{3} + x \right) \) and \( \cos(A-B) = \cos(A)\cos(B) + \sin(A)\sin(B) \) to \( \cos\left( \frac{\pi}{3} - x \right) \). After substitution, the expression will look like this: \( \cos\left( \frac{\pi}{3} \right)\cos(x) - \sin\left( \frac{\pi}{3} \right)\sin(x) + \cos\left( \frac{\pi}{3} \right)\cos(x) + \sin\left( \frac{\pi}{3} \right)\sin(x) \).
02

Simplify the Expression

Now, simplify the expression obtained above: \( \cos\left( \frac{\pi}{3} \right)\cos(x) - \sin\left( \frac{\pi}{3} \right)\sin(x) + \cos\left( \frac{\pi}{3} \right)\cos(x) + \sin\left( \frac{\pi}{3} \right)\sin(x) \). Because plus and minus of the same terms cancel each other, we are left with \( 2\cos\left( \frac{\pi}{3} \right)\cos(x) \).
03

Find the Value and Substitute

Next, find the value of \( \cos\left( \frac{\pi}{3} \right) \) which is \( \frac{1}{2} \). Substituting this value into the expression, we get \( 2 * \frac{1}{2} * \cos(x) \) which simplifies to \( \cos(x) \) confirming the given identity.

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