Question:

Step 1: Simplify the numerator of the fraction inside the logarithm. We start with the numerator: $\sin^2 A + \cos^2 A + \tan^2 A - \sec^2 A \cdot \sin^2 A$ Using the fundamental identity $\sin^2 A + \cos^2 A = 1$: $$1 + \tan^2 A - \sec^2 A \cdot \sin^2 A$$ Recall that $1 + \tan^2 A = \sec^2 A$, so: $$\sec^2 A - \sec^2 A \cdot \sin^2 A$$ Factor out $\sec^2 A$: $$\sec^2 A(1 - \sin^2 A)$$ Using the identity $1 - \sin^2 A = \cos^2 A$: $$\sec^2 A \cdot \cos^2 A$$ Since $\sec A = \frac{1}{\cos A}$, we have $\sec^2 A \cdot \cos^2 A = \frac{1}{\cos^2 A} \cdot \cos^2 A = 1$ Step 2: Simplify the denominator of the fraction inside the logarithm. The denominator is: $(1 + \tan^2 A)(1 - \sin^2 A)$ Using the identity $1 + \tan^2 A = \sec^2 A$: $$\sec^2 A(1 - \sin^2 A)$$ Using the identity $1 - \sin^2 A = \cos^2 A$: $$\sec^2 A \cdot \cos^2 A = 1$$ Step 3: Evaluate the fraction. Now we can compute: $$\frac{\text{Numerator}}{\text{Denominator}} = \frac{1}{1} = 1$$ Step 4: Evaluate the logarithm. $$\log_5(1) = 0$$ This is because $5^0 = 1$ by definition of logarithms. Step 5: Evaluate the cosine of the result. $$\cos(0) = 1$$ **Final Answer:** The value of the given expression is $\boxed{1}$, which corresponds to **Option 4**.