Draw the right-angled triangle and determine the third side.

sin A = $\frac{3}{5}$

sin(θ) = $\frac{\mathrm{Opposite}}{\mathrm{Hypothenuse}}$

In sin A = $\frac{3}{5}$, the opposite side of A is 3 and hypothenuse side is 5.

From pythagoras theorem

The longest side (side opposite/facing the right-angle) is the hypothenuse (c)

From the triangle, the side facing the right-angle is y, hence y is the hypothenuse (c)

b² + 3² = 5²

b² + 9 = 25

b² = 25 - 9

b² = 16

Take square root of both sides.

b = $\sqrt{\mathrm{16}}$ = 4

TOA

tan(θ) = $\frac{\mathrm{Opposite}}{\mathrm{Adjacent}}$

Side opposite to A is 3 and adjacent to A is 4

tan A = $\frac{3}{4}$

CAH

cos(θ) = $\frac{\mathrm{Adjacent}}{\mathrm{Hypothenuse}}$

Hypothenuse side of A is 5

cos A = $\frac{4}{5}$

tan A - cos A = $\frac{3}{4}$ - $\frac{4}{5}$

tan A - cos A = $\frac{\mathrm{5\; x\; 3\; -\; 4\; x\; 4}}{20}$

tan A - cos A = $\frac{\mathrm{15\; -\; 16}}{20}$

tan A - cos A = -$\frac{1}{20}$