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Chapter 5: Problem 124

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

Short Answer

Expert verified
Yes, the given identity is verified, as we ended up with \(\cos x = \cos x\), after applying the cosine addition and subtraction formulae and simplifying.

Step by step solution

01

Apply the Cosine Addition and Subtraction Formulae

If you recall the cosine addition and subtraction formulae, they are: \( \cos(a+b) = \cos a . \cos b - \sin a . \sin b\) and \( \cos(a-b) = \cos a . \cos b + \sin a . \sin b \). Now, apply these to the given problem. \[ \cos \left( \frac{\pi}{3} + x \right) + \cos \left( \frac{\pi}{3} - x \right) \] becomes \[ \left( \cos \frac{\pi}{3} . \cos x - \sin \frac{\pi}{3} . \sin x \right) + \left( \cos \ rac{\pi}{3} . \cos x + \sin \frac{\pi}{3} . \sin x \right)\]
02

Simplify the Equation

You can simplify the equation by combining like terms: \[ 2 \cos \frac{\pi}{3} . \cos x = \sqrt{3} \cos x \]. The factor of 2 and \sqrt{3} can be eliminated because they are common on both sides, so you get \(\cos x = \cos x\). This verifies the given identity.

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