We are given a trigonometric expression.

tan (9 0^{\circ} - A)

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

9 0^{\circ} .

A + B = 9 0^{\circ}

We can rewrite this sum as a difference that will allow us to rewrite the given expression.

9 0^{\circ} - A = B ⇓ tan (9 0^{\circ} - A) = tan B

We want to derive a cofunction identity for the given expression. To do so, we will start by drawing a right triangle

A B C .

Now, let's recall the trigonometric ratios for both tangent and its reciprocal ratio, cotangent.

tan θ = \frac{Opposite}{Adjacent} cot θ = \frac{Adjacent}{Opposite}

Since we have

tan B,

we need the opposite and adjacent sides to

∠ 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.

tan B = \frac{Opposite}{Adjacent} = \frac{b}{a}

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

∠ A .

Finally, we can write the cofunction identity for the given expression.

tan (9 0^{\circ} - A) = cot A

Exercise