To prove the trigonometric identity $\cot(x) + \tan(x) = \csc(x) \sec(x)$, let's follow the steps shown in the image and fill in the missing parts:

Step 1: Rewrite cotangent and tangent in terms of sine and cosine:

$\cot(x) + \tan(x) = \frac{\cos(x)}{\sin(x)} + \frac{\sin(x)}{\cos(x)}$

Step 2: Combine the fractions by finding a common denominator:

$\frac{\cos(x)}{\sin(x)} + \frac{\sin(x)}{\cos(x)} = \frac{\cos^2(x) + \sin^2(x)}{\sin(x) \cos(x)}$

Step 3: Use the Pythagorean identity $\cos^2(x) + \sin^2(x) = 1$:

$\frac{\cos^2(x) + \sin^2(x)}{\sin(x) \cos(x)} = \frac{1}{\sin(x) \cos(x)}$

Step 4: Rewrite the expression using reciprocal identities:

$\frac{1}{\sin(x) \cos(x)} = \left(\frac{1}{\sin(x)}\right) \left(\frac{1}{\cos(x)}\right) = \csc(x) \sec(x)$

Therefore, we have proven the identity:

$\cot(x) + \tan(x) = \csc(x) \sec(x).$

Would you like more details or have any questions?

Related Questions:

1. How can we prove other trigonometric identities using similar methods?
2. What are the key trigonometric identities that are essential for calculus?
3. How do reciprocal identities help simplify complex trigonometric expressions?
4. Can this method be used to verify the identity $\sin(x) + \cos(x) = \sqrt{2} \sin(x + \frac{\pi}{4})$?
5. What is the significance of the Pythagorean identity in trigonometric proofs?

Tip:

When proving trigonometric identities, always look for opportunities to use fundamental identities like reciprocal, Pythagorean, or quotient identities to simplify expressions.