<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 trigonometric expression inside the logarithm using fundamental identities (sin²A + cos²A = 1, sec²A = 1 + tan²A), then recognize that log₅(1) = 0, making cos(0) = 1.
<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, so sec²A·sin²A = (1 + tan²A)sin²A:</p><p>Numerator = 1 + tan²A - (1 + tan²A)sin²A = (1 + tan²A)(1 - sin²A)</p><p><strong>Step 3:</strong> Simplify the denominator:</p><p>(1 + tan²A)(1 - sin²A) = (1 + tan²A)cos²A</p><p><strong>Step 4:</strong> Form the fraction:</p><p>$$\frac{(1 + tan^2A)(1 - sin^2A)}{(1 + tan^2A)(1 - sin^2A)} = 1$$</p><p><strong>Step 5:</strong> Evaluate the logarithm and cosine:</p><p>cos[log₅(1)] = cos(0) = 1</p><p>∴ Answer: D (which equals 1)</p>
Correct Answer: D