To prove the identity
$\tan\left(\frac{\pi}{2} - u\right) = \cot u$
using trigonometric addition and subtraction formulas, follow the steps below:

Proof:

1. Recall the tangent subtraction formula:
   $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}.$

2. Substitute $A = \frac{\pi}{2}$ and $B = u$:
   $\tan\left(\frac{\pi}{2} - u\right) = \frac{\tan\left(\frac{\pi}{2}\right) - \tan(u)}{1 + \tan\left(\frac{\pi}{2}\right)\tan(u)}.$

3. Evaluate $\tan\left(\frac{\pi}{2}\right)$:
   From trigonometric properties,
   $\tan\left(\frac{\pi}{2}\right) = \text{undefined (approaches infinity)}.$
   Instead of using the tangent formula explicitly, rely on the cofunction identity for sine and cosine: $\tan\left(\frac{\pi}{2} - u\right) = \frac{\sin\left(\frac{\pi}{2} - u\right)}{\cos\left(\frac{\pi}{2} - u\right)}.$

4. Use cofunction identities:

   • $\sin\left(\frac{\pi}{2} - u\right) = \cos u$,
   • $\cos\left(\frac{\pi}{2} - u\right) = \sin u$.

   Therefore,
   $\tan\left(\frac{\pi}{2} - u\right) = \frac{\cos u}{\sin u}.$

5. Simplify:
   By definition of cotangent:
   $\cot u = \frac{\cos u}{\sin u}.$

6. Conclusion:
   $\tan\left(\frac{\pi}{2} - u\right) = \cot u.$
   The identity is proven.

Summary:

The identity holds because the tangent and cotangent functions are cofunctions, and the cofunction relationships for sine and cosine simplify the expression.

Do you need further clarification on any of the steps? Let me know!

Related Questions:

1. How can you prove the cofunction identity for $\sin\left(\frac{\pi}{2} - u\right) = \cos u$?
2. What is the cofunction identity for $\cos\left(\frac{\pi}{2} - u\right)$?
3. How do you derive the tangent addition formula?
4. What are the values of $\tan$ and $\cot$ in different quadrants of the unit circle?
5. How does the cofunction identity relate to complementary angles?

Tip:

Cofunction identities are based on the idea that the angles sum to $\frac{\pi}{2}$ (or $90^\circ$), which reflects symmetry on the unit circle.