Question

find the exact value of the expression

$\cos \left(-\frac{5\mathrm{\pi }}{4}\right)-\cos \left(\frac{3\mathrm{\pi }}{4}\right)$

Short answer

Value of the expression is 0

Step-by-step solution

Step 1. Given information

$\cos \left(-\frac{5\mathrm{\pi }}{4}\right)-\cos \left(\frac{3\mathrm{\pi }}{4}\right)$

Step 2. Rewriting the expression

As cos is a even function so, cos(x)=cos(-x)

So given expression can be written as$\cos \left(\frac{5\mathrm{\pi }}{4}\right)-\cos \left(\frac{3\mathrm{\pi }}{4}\right)$

Step 3. Calculations

$$\begin{gathered} Weknowthat \\ \cos \left(\mathrm{\pi }+\mathrm{x}\right)=-\cos \left(x\right) \\ \cos \left(\mathrm{\pi }-\mathrm{x}\right)=-\cos \left(x\right) \\ So, \\ \cos \left(\frac{5\mathrm{\pi }}{4}\right)=\cos (\mathrm{\pi }+\frac{\mathrm{\pi }}{4})=-\cos \left(\frac{\mathrm{\pi }}{4}\right) \\ \cos \left(\frac{53\mathrm{\pi }}{4}\right)=\cos (\mathrm{\pi }-\frac{\mathrm{\pi }}{4})=-\cos \left(\frac{\mathrm{\pi }}{4}\right) \\ \mathrm{So},\cos \left(\frac{5\mathrm{\pi }}{4}\right)-\cos \left(\frac{53\mathrm{\pi }}{4}\right)=0 \end{gathered}$$