If cosecx = 5over 4 , then the value of 5sin²x\ +\ 8cot²x over 27tan²x\ -\ 125cos²x is:

If cosecx = 5over 4 , then the value of 5sin²x\ +\ 8cot²x…

If cosecx = $5\over 4$ , then the value of $5sin^2x\ +\ 8cot^2x \over 27\tan^2x\ -\ 125cos^2x$ is:

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Correct answer: B

Explanation

The correct answer is option 2 i.e. $77\over 30$ .

cosec x = 5/4

sin x = 1/cosec x = 4/5

cos²x = 1 - sin²x = 1 - (4/5)² = 1 - 16/25 = 9/25

cos x = √(9/25) = 3/5

tan x = sin x/cos x = (4/5)/(3/5) = 4/3

cot x = 1/tan x = 3/4

Now, (5sin²x + 8cot²x)/(27tan²x - 125cos²x)

= [5(4/5)² + 8(3/4)²]/[27(4/3)² - 125(3/5)²]

= [5(16/25) + 8(9/16)]/[27(16/9) - 125(9/25)]

= (16/5 + 9/2)/(48 - 45)

= (77/10)/3

= 77/30

Therefore, the required value is 77/30.