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7-113.

Given $\sec A = \frac{5}{3}$ and $\cot B = \frac { 12 } { 5 }$ where $0 \lt A$ , $B < \frac { \pi } { 2 }$ , find the exact value of each trig expression.

$\sin A$
Step 1(a):
Draw a reference triangle in Quadrant 1 for both angles.
Step 2(a):
Label the 3rd side.
Both are pythagorean triples: $3,4,5$ and $5,12,13$
Step 3(a):
Find sin A from the triangle sides.
Answer (a):
$\frac{4}{5}$

$\cos B$
Hint (b):
Find $\cos B$ from the triangle sides.
Answer (b):
$\frac{12}{13}$

$\sin\left(A + B\right)$
Hint (c):
Use the sum and difference formula:
sin(A + B) = sinAcosB + cosAsinB
$\text{sin}(A+B)=\frac{4}{5}\cdot \frac{12}{13}+\frac{3}{5}\cdot \frac{5}{13}=$
Answer (c):
$\frac{63}{65}$

$\cos\left(A + B\right)$
Hint (d):
Use the sum and difference formula:
$\cos\left(A + B\right) = \cos A \cos B − \sin A \sin B$

$\tan\left(A + B\right)$
Hint (e):
$\text{tan}(A+B)=\frac{\text{sin}(A+B)}{\text{cos}(A+B)}$

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