Let's break down and solve the problem in Question number 3 step-by-step.

Problem:

We are given: $5 \cos^2(\theta) - 6 \sec^2(\theta) + 5 \sin^2(\theta) + 6 \tan^2(\theta)$ We need to simplify this expression and find its equivalent.

Step-by-step simplification:

1. Start with the given expression: $5 \cos^2(\theta) - 6 \sec^2(\theta) + 5 \sin^2(\theta) + 6 \tan^2(\theta)$

2. Use trigonometric identities:

   • $\sec^2(\theta) = 1 + \tan^2(\theta)$
   • $\sin^2(\theta) + \cos^2(\theta) = 1$

3. Substitute $\sec^2(\theta)$ with $1 + \tan^2(\theta)$: $5 \cos^2(\theta) - 6 (1 + \tan^2(\theta)) + 5 \sin^2(\theta) + 6 \tan^2(\theta)$

4. Expand the expression: $5 \cos^2(\theta) - 6 - 6 \tan^2(\theta) + 5 \sin^2(\theta) + 6 \tan^2(\theta)$

5. Combine like terms:

   • The $6 \tan^2(\theta)$ terms cancel out. $5 \cos^2(\theta) + 5 \sin^2(\theta) - 6$

6. Use the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$: $5(1) - 6 = 5 - 6 = -1$

Conclusion:

The simplified expression is: $-1$

So the correct answer to Question number 3 is: $\boxed{-1}$

Would you like details or further explanations on any of these steps?

Here are 5 related questions for further practice:

1. Simplify $3 \sin^2(\theta) + 4 \cos^2(\theta)$.
2. What is the value of $1 + \tan^2(\theta)$?
3. How do you prove the identity $\sec^2(\theta) - \tan^2(\theta) = 1$?
4. Simplify $\sin^4(\theta) + \cos^4(\theta)$.
5. Solve for $\theta$ when $\sec^2(\theta) = 4$.

Tip: Remember, Pythagorean identities like $\sin^2(\theta) + \cos^2(\theta) = 1$ are fundamental in trigonometry and help simplify many expressions.