To solve the expression $\frac{\sin^2 \theta}{\cos^2 \theta} - 1 - \frac{\cos^4 \theta}{\cos^2 \theta}$, let's break it down step by step:

Step 1: Simplify the first term

The first term is $\frac{\sin^2 \theta}{\cos^2 \theta}$. This can be rewritten using the identity for tangent:

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

Step 2: Simplify the second term

The second term is $1$, which remains unchanged.

Step 3: Simplify the third term

The third term is $\frac{\cos^4 \theta}{\cos^2 \theta}$. This can be simplified by canceling out $\cos^2 \theta$ in the numerator and the denominator:

$\frac{\cos^4 \theta}{\cos^2 \theta} = \cos^2 \theta$

Step 4: Combine the simplified terms

Now, substitute the simplified expressions back into the original equation:

$\tan^2 \theta - 1 - \cos^2 \theta$

Step 5: Apply a trigonometric identity

We can use the trigonometric identity $\tan^2 \theta = \sec^2 \theta - 1$ to further simplify:

$(\sec^2 \theta - 1) - 1 - \cos^2 \theta$

Simplify further:

$\sec^2 \theta - 2 - \cos^2 \theta$

Final Expression

The simplified expression is:

$\sec^2 \theta - \cos^2 \theta - 2$

Would you like further details on this solution, or do you have any questions?

5 Related Questions:

1. How is the identity $\tan^2 \theta = \sec^2 \theta - 1$ derived?
2. What are other useful trigonometric identities for simplifying expressions?
3. How can we interpret $\sec^2 \theta - \cos^2 \theta$ geometrically?
4. What are the steps to simplify more complex trigonometric expressions?
5. How does this trigonometric expression relate to others involving sine and cosine?

Tip:

Always look for opportunities to apply trigonometric identities, as they can significantly simplify complex expressions.