We are given a trigonometric expression.
csc (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
⇓
csc (90^(∘) - A ) = csc 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 cosecant and for secant.
csc θ=Hypotenuse/Opposite
sec θ=Hypotenuse/Adjacent
Since we have csc B, we need the length of the hypotenuse and the length of the opposite side to ∠ B. From the diagram, we can see that b is the opposite side to B and c is the hypotenuse. We will substitute these values into the expression for cosecant ratio.
csc B = Hypotenuse/Opposite=c/b
We can rewrite this equation using secant by noting that c is the length of the hypotenuse and b is the length of the adjacent side angle to A. Then, we will have the quotient between the hypotenuse and the adjacent side to ∠ A.
Finally, we can write the cofunction identity for the given expression.
csc (90^(∘) - A) = sec A
