The problem presented in the image shows two equations involving trigonometric functions:

1. $\sec\theta + \tan\theta = m$
2. $\sec\theta - \tan\theta = n$

We are tasked to prove that:

$m^2 - n^2 = 4\sqrt{mn}$

Step 1: Use the identity for the difference of squares

We start by applying the identity $m^2 - n^2 = (m - n)(m + n)$:

$m^2 - n^2 = (\sec\theta + \tan\theta + \sec\theta - \tan\theta)(\sec\theta + \tan\theta - (\sec\theta - \tan\theta))$

This simplifies to:

$m^2 - n^2 = (2\sec\theta)(2\tan\theta)$

Thus,

$m^2 - n^2 = 4\sec\theta \cdot \tan\theta$

Step 2: Use the identities for secant and tangent

We know from trigonometric identities that:

$\sec\theta = \frac{1}{\cos\theta} \quad \text{and} \quad \tan\theta = \frac{\sin\theta}{\cos\theta}$

Therefore, $\sec\theta \cdot \tan\theta$ becomes:

$\sec\theta \cdot \tan\theta = \frac{1}{\cos\theta} \cdot \frac{\sin\theta}{\cos\theta} = \frac{\sin\theta}{\cos^2\theta}$

Thus:

$m^2 - n^2 = 4 \cdot \frac{\sin\theta}{\cos^2\theta}$

Step 3: Express $\sqrt{mn}$

From the given equations $m = \sec\theta + \tan\theta$ and $n = \sec\theta - \tan\theta$, the product $mn$ becomes:

$mn = (\sec\theta + \tan\theta)(\sec\theta - \tan\theta) = \sec^2\theta - \tan^2\theta$

Using the Pythagorean identity $\sec^2\theta - \tan^2\theta = 1$, we have:

$mn = 1$

Therefore:

$\sqrt{mn} = \sqrt{1} = 1$

Step 4: Final equation

Substituting into the equation $m^2 - n^2 = 4\sqrt{mn}$, we get:

$m^2 - n^2 = 4\cdot 1 = 4$

Thus, the equation holds true.

Let me know if you'd like more details on any of the steps, or if you have further questions.

Related questions:

1. Can you prove the identity $\sec^2\theta - \tan^2\theta = 1$ using basic trigonometric identities?
2. What are the other forms of trigonometric identities for secant and tangent functions?
3. How can we derive the formula for $\sec\theta$ and $\tan\theta$ in terms of sine and cosine?
4. Can you solve for $\theta$ in terms of $m$ and $n$ from the given equations?
5. How can these trigonometric equations be applied in real-world scenarios?

Tip:

When dealing with trigonometric identities, always keep basic identities like the Pythagorean identities handy. They often simplify complex expressions.