To prove the cofunction identity

$\sec\left(\frac{\pi}{2} - u\right) = \csc(u)$

using the reciprocal and subtraction identities, we will follow these steps:

Step 1: Rewrite $\sec\left(\frac{\pi}{2} - u\right)$ using the Reciprocal Identity

The secant function is defined as the reciprocal of the cosine function, so we can write:

$\sec\left(\frac{\pi}{2} - u\right) = \frac{1}{\cos\left(\frac{\pi}{2} - u\right)}$

Step 2: Apply the Cofunction Identity for Cosine

Using the subtraction formula, we know that:

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

This follows from the cofunction identity, which states that $\cos\left(\frac{\pi}{2} - x\right) = \sin(x)$ for any angle $x$.

Thus,

$\sec\left(\frac{\pi}{2} - u\right) = \frac{1}{\sin(u)}$

Step 3: Recognize the Result as the Definition of Cosecant

By definition, $\csc(u) = \frac{1}{\sin(u)}$. Therefore,

$\sec\left(\frac{\pi}{2} - u\right) = \csc(u)$

Conclusion

We have shown that:

$\sec\left(\frac{\pi}{2} - u\right) = \csc(u)$

This completes the proof using the reciprocal identity and the subtraction formula.

Would you like a breakdown of each step or have any questions? Here are a few related questions for deeper understanding:

1. What are other examples of cofunction identities in trigonometry?
2. How do the reciprocal identities relate to cofunction identities?
3. Can you prove a similar identity for $\sin\left(\frac{\pi}{2} - u\right) = \cos(u)$?
4. How are addition and subtraction formulas generally applied in trigonometric proofs?
5. What role does the unit circle play in understanding these identities?

Tip: Remember that cofunction identities relate trigonometric functions of complementary angles.