We are given a trigonometric expression.
cot (90^(∘) - A)
We will 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
⇓
cot (90^(∘) - A) = cot 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.
tan θ=Opposite/Adjacent
cot θ=Adjacent/Opposite
Since we have cot B, we need the length of the opposite and adjacent side to ∠B. From the diagram, we can see that b is the opposite side to B and a is their adjacent side. We will substitute these values into the expression for cotangent ratio.
cot B = Adjacent/Opposite=a/b
We can rewrite this equation using tangent 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 opposite and the adjacent side to ∠A.
Finally, we can write the cofunction identity for the given expression.
cot (90^(∘) - A) = tan A
