If cosecA + cotA = a √(b) , then find the value of ((a²b - 1))/((a²b + 1)) .

If cosecA + cotA = a √(b) , then find the value of ((a²b…

If cosecA + cotA = a $\sqrt{b}$ , then find the value of $\frac{(a^2b - 1)}{(a^2b + 1)}$ .

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Correct answer: A

Explanation

The correct answer is option 1 i.e. cosA.

cosec A + cot A = a√b.

Square both sides: (cosec A + cot A)² = (a√b)²

Expanding: cosec² A + 2(cosec A)(cot A) + cot² A = a²b

Use identity: cosec² A = 1 + cot² A

Substituting: (1 + cot² A) + 2(cosec A)(cot A) + cot² A = a²b

Simplifying: 1 + 2cot² A + 2(cosec A)(cot A) = a²b

We need to find: (a²b − 1) / (a²b + 1)

Substitute a²b = 1 + 2cot² A + 2(cosec A)(cot A) into the expression.

Simplifying the numerator and denominator:

= (2cot² A + 2(cosec A)(cot A)) / (2cot² A + 2(cosec A)(cot A) + 2)

Factor out 2 and cancel:

= (cot² A + (cosec A)(cot A)) / (cot² A + (cosec A)(cot A) + 1)

= cot A / cosec A

= cos A