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Chapter 5: Problem 65

Prove the identity. $$\cos (\pi-\theta)+\sin \left(\frac{\pi}{2}+\theta\right)=0$$

Short Answer

Expert verified
By replacing the initial expressions with their respective identities and simplifying the resulting expression, it is seen that the equation \(\cos (\pi-\theta)+\sin \left(\frac{\pi}{2}+\theta\right)\) does indeed equal zero.

Step by step solution

01

Applying the first identity

The identity for \(\cos(\pi - \theta)\) is \(-\cos(\theta)\). So, replace \(\cos(\pi - \theta)\) with \(-\cos(\theta)\) in the equation.
02

Applying the second identity

Next, apply the identity for \(\sin\left(\frac{\pi}{2} + \theta\right)\), which is \(\cos(\theta)\). Now replace \(\sin\left(\frac{\pi}{2} + \theta\right)\) with \(\cos(\theta)\) in the equation.
03

Simplifying the expression

Now simplify the resulting expression \(-\cos(\theta) + \cos(\theta)\) . The two terms will cancel each other out, giving zero.

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Most popular questions from this chapter

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