Question

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that shows it is false. $$ \begin{aligned} &\text { The graph of } y=\cos \left(x+\frac{\pi}{4}\right) \text { is the same as the graph of }\\\ &y=-\sin \left(x-\frac{\pi}{4}\right) \end{aligned} $$

Short answer

The statement is false. The graph of \(y= \cos\left(x + \frac{\pi}{4}\right)\) is not the same as the graph of \(y = -\sin\left(x - \frac{\pi}{4}\right)\), as their expressions are not identical and no trigonometric identity was sufficient to establish their equivalence.

Step-by-step solution

Determine the cosine of the first function

The cosine of the first function is already given by the expression: \[y_{1} = \cos \left(x + \frac{\pi}{4}\right)\]

Determine the sine of the second function

The sine of the second function is given by the expression: \[y_{2} = -\sin \left(x - \frac{\pi}{4}\right)\]

Use a trigonometric identity to convert sine to cosine

We can use the trigonometric identity: \[\sin \left(\frac{\pi}{2} - \theta\right) = \cos(\theta)\] Now, we can rewrite \(y_{2}\) utilizing this identity: \[y_{2} = -\sin \left(\frac{\pi}{2} - \left(x + \frac{\pi}{4}\right)\right)\]

Simplify the expression

Now we can simplify this expression for \(y_{2}\): \[y_{2} = -\sin \left(\frac{\pi}{2} - x - \frac{\pi}{4}\right)\] \[y_{2} = -\sin \left(\frac{\pi}{4} - x\right)\]

Compare the expressions

We will compare the expressions for \(y_{1}\) and \(y_{2}\): \[y_{1} = \cos \left(x + \frac{\pi}{4}\right)\] \[y_{2} = -\sin \left(\frac{\pi}{4} - x\right)\] The trigonometric identity we used is not sufficient to establish equivalence between the expressions. Furthermore, the two expressions are not identical, and their graphs are not the same.

Conclusion

The statement is false. The graph of \(y= \cos\left(x + \frac{\pi}{4}\right)\) is not the same as the graph of \(y = -\sin\left(x - \frac{\pi}{4}\right)\).