We are given a trigonometric expression.
tan (90^(∘) - 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 90 ^(∘).
A+ B = 90^(∘)
We can rewrite this sum as a difference that will allow us to rewrite the given expression.
90^(∘)- A= B
⇓
tan (90^(∘) - 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 ABC.
Now, let's recall the trigonometric ratios for both tangent and its reciprocal ratio, cotangent.
cc
tan θ=Opposite/Adjacent & cot θ=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 = Opposite/Adjacent=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 (90^(∘) - A) = cot A
