find the exact value of $\\cos\\left(-\\frac{5\\pi}{4}\\right)$.\n$\\cos\\left(-\\frac{5\\pi}{4}\\right)=\\sq…

Explanation:

Step1: Use the cosine even - function property

The cosine function is even, i.e., (\cos(-x)=\cos(x)). So, (\cos\left(-\frac{5\pi}{4}\right)=\cos\left(\frac{5\pi}{4}\right)).

Step2: Rewrite the angle

We can rewrite (\frac{5\pi}{4}) as (\pi+\frac{\pi}{4}). Then, by the cosine of a sum formula (\cos(A + B)=\cos A\cos B-\sin A\sin B) (where (A=\pi) and (B = \frac{\pi}{4})), (\cos\left(\pi+\frac{\pi}{4}\right)=\cos\pi\cos\frac{\pi}{4}-\sin\pi\sin\frac{\pi}{4}). Since (\cos\pi=- 1), (\sin\pi = 0), and (\cos\frac{\pi}{4}=\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}), we have (\cos\left(\pi+\frac{\pi}{4}\right)=(-1)\times\frac{\sqrt{2}}{2}-0\times\frac{\sqrt{2}}{2}).

Step3: Calculate the value

(\cos\left(\pi+\frac{\pi}{4}\right)=-\frac{\sqrt{2}}{2}).

Answer:

(-\frac{\sqrt{2}}{2})