Exercise 13 Page 949

We are given a trigonometric expression. Consider that $A$ is an acute angle of a right triangle. The acute angles of a right triangle are complementary, which means that their sum is $90^{\circ }.$ We can rewrite this sum as a difference that will allow us to rewrite the given expression. We want to derive a cofunction identity for the given expression. To do so, we will start by drawing a right triangle $ABC.$ Now, let's recall the trigonometric ratios for both tangent and its reciprocal ratio, cotangent. Since we have $\tan B,$ we need the opposite and adjacent sides to $\angle B.$ From the diagram, we can see that $b$ is the opposite side to $B$ and $a$ is its adjacent side. We will substitute these values into the expression for tangent ratio. We can rewrite this equation using cotangent by noting that $b$ is the length of the adjacent side and $a$ is the length of the opposite side angle to $A.$ Then, we will have the quotient between the adjacent and the opposite side to $\angle A.$ Finally, we can write the cofunction identity for the given expression.