if θ = 7π/4, numerically show the following e is an identity using unit - circle values. cos(7π/4)+sin(7π/4)t…

Explanation:

Step1: Find cosine value

Using unit - circle, $\cos(\frac{7\pi}{4})=\frac{\sqrt{2}}{2}$

Step2: Find sine value

Using unit - circle, $\sin(\frac{7\pi}{4})=-\frac{\sqrt{2}}{2}$

Step3: Find tangent value

$\tan(\frac{7\pi}{4})=\frac{\sin(\frac{7\pi}{4})}{\cos(\frac{7\pi}{4})}=\frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}=- 1$

Step4: Find secant value

$\sec(\frac{7\pi}{4})=\frac{1}{\cos(\frac{7\pi}{4})}=\sqrt{2}$

Step5: Evaluate left - hand side

$\cos(\frac{7\pi}{4})+\sin(\frac{7\pi}{4})\tan(\frac{7\pi}{4})=\frac{\sqrt{2}}{2}+(-\frac{\sqrt{2}}{2})\times(-1)$ $=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}=\sqrt{2}$

Step6: Compare

The left - hand side $\sqrt{2}$ is equal to the right - hand side $\sqrt{2}$

Answer:

The given equation is an identity since both sides evaluate to $\sqrt{2}$.