watch the video and then solve the problem given below.\nclick here to watch the video.\nuse the cosine of a…

Explanation:

Step1: Find $\cos s$ and $\cos t$

Using the Pythagorean identity $\sin^{2}\alpha+\cos^{2}\alpha = 1$. For $s$: Since $\sin s=\frac{5}{13}$ and $s$ is in quadrant II, then $\cos s=-\sqrt{1 - \sin^{2}s}=-\sqrt{1-\left(\frac{5}{13}\right)^{2}}=-\sqrt{\frac{169 - 25}{169}}=-\frac{12}{13}$. For $t$: Since $\sin t=-\frac{4}{5}$ and $t$ is in quadrant III, then $\cos t=-\sqrt{1-\sin^{2}t}=-\sqrt{1-\left(-\frac{4}{5}\right)^{2}}=-\sqrt{\frac{25 - 16}{25}}=-\frac{3}{5}$.

Step2: Use the cosine - of - a - sum formula

The formula for $\cos(A + B)=\cos A\cos B-\sin A\sin B$. Substitute $A = s$ and $B = t$: $\cos(s + t)=\cos s\cos t-\sin s\sin t$. Substitute $\cos s=-\frac{12}{13},\cos t=-\frac{3}{5},\sin s=\frac{5}{13},\sin t=-\frac{4}{5}$ into the formula: [ \begin{align*} \cos(s + t)&=\left(-\frac{12}{13}\right)\times\left(-\frac{3}{5}\right)-\frac{5}{13}\times\left(-\frac{4}{5}\right)\ &=\frac{36}{65}+\frac{20}{65}\ &=\frac{36 + 20}{65}\ &=\frac{56}{65} \end{align*} ]

Answer:

$\frac{56}{65}$