Our friend claims that it is possible to use the difference formula for tangent to derive the following cofunction identity.
tan ( π/2-θ)= cot θ
To see if this claim is true, we will start by recalling the difference formula for tangent.
tan ( a- b) = tan a -tan b/1+tan a tan bNow we will apply this formula into the left-hand side of the cofunction identity.
To continue simplifying, we need to find the value of tan π2. To do so, we will recall the trigonometric function for tangent.
tan θ = y/x
Now, we will use the unit circle to evaluate this function in θ= π2.
Let's start by drawing the angle in standard position and paying attention to the point of intersection of the circle and the terminal side of the angle.
We can see that the point of intersection of the terminal side of the angle π2 and the unit circle is ( 0, 1). With this information we can evaluate the trigonometric function for tangent.
tan θ = y/x ⇒ tan π2 = 1/0
Remember that the denominator of a fraction can never be equal to 0, since the division by 0 is always undefined. This means that we cannot use the difference formula to derive the cofunction identity because the value of tan π2 is undefined.
tan π2 -tan θ/1+tan π2 tan θ ≠cot θ
Therefore, the claim of our friend is not correct.
