If m = a secA and y = b tanA, then find the value of b²m² - a²y²+ a² y²over b² m² + cos²A.

If m = a secA and y = b tanA, then find the value of b²m²…

If m = a secA and y = b tanA, then find the value of b²m² - a²y²+ $a^2 y^2\over b^2 m^2$ + cos²A.

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Correct answer: D

Explanation

The correct answer is option 4 i.e. a² b² +1.

m = a secA ⇒ m² = a²sec²A

y = b tanA ⇒ y² = b²tan²A

So, b²m² - a²y²+ (a²y²/b²m²) + cos²A

= b²(a²sec²A) - a²(b²tan²A) + [a²(b²tan²A)/b²(a²sec²A)] + cos²A

= a²b²sec²A - a²b²tan²A + (a²b²tan²A/a²b²sec²A) + cos²A

= a²b²sec²A - a²b²tan²A + (tan²A/sec²A) + cos²A

= a²b²(sec²A - tan²A) + (tanA/secA)² + cos²A

= a²b² + (tanA/secA)² + cos²A [∵ sec²A - tan²A = 1]

$=a^2b^2+\left(\frac{cosA/sinA}{1/cosA}\right)^2+cos^2A$ [∵ tanA = cosA/sinA and secA = 1/cosA]

= a²b² + sin²A + cos²A

= a²b² + 1 [∵ sin²A + cos²A = 1]

Therefore, the value of the expression is a²b² + 1