<p><strong>172.</strong> The value of \(\cos\left[\log_5\left(\dfrac{\sin^2 A + \cos^2 A + \tan^2 A - \sec^2 A \cdot \sin^2 A}{(1 + \tan^2 A)(1 - \sin^2 A)}\right)\right]\) is equal to:</p>
Step-by-Step Solution
Key Concept: Simplify the argument of the logarithm using fundamental trigonometric identities: sin²A + cos²A = 1, sec²A = 1 + tan²A, and 1 - sin²A = cos²A. This reduces the complex fraction to a simple number whose logarithm can be evaluated.
<p><strong>Step 1:</strong> Simplify the numerator using sin²A + cos²A = 1:</p><p>sin²A + cos²A + tan²A - sec²A·sin²A = 1 + tan²A - sec²A·sin²A</p><p><strong>Step 2:</strong> Use sec²A = 1 + tan²A to rewrite sec²A·sin²A:</p><p>sec²A·sin²A = (1 + tan²A)sin²A</p><p><strong>Step 3:</strong> Substitute back into the numerator:</p><p>1 + tan²A - (1 + tan²A)sin²A = (1 + tan²A)(1 - sin²A)</p><p><strong>Step 4:</strong> Write the full fraction with denominator (1 + tan²A)(1 - sin²A):</p><p>$$\frac{(1 + \tan^2 A)(1 - \sin^2 A)}{(1 + \tan^2 A)(1 - \sin^2 A)} = 1$$</p><p><strong>Step 5:</strong> Evaluate the logarithm and cosine:</p><p>$$\cos[\log_5(1)] = \cos(0) = 1$$</p><p>∴ Answer: D</p>
Correct Answer: D