Final answer:
To find the value of cot 9, we can use the identity: cot^2x + 1 = csc^2x. First, solve for csc^2 9. Since cosec 9 = 5/4, we can find the value of sin 9 using the reciprocal property. Then, substitute the value of sin 9 into the identity to solve for cot 9.
Step-by-step explanation:
To find the value of cot 9, we can use the identity: cot^2x + 1 = csc^2x. Since cosec 9 = 5/4, we can substitute it in the equation to solve for cot 9.
First, solve for csc^2 9:
csc^2 9 = (1/sin^2 9)
Since we know the value of cosec 9, we can find the value of sin 9 using the reciprocal property: cosec x = 1/sin x.
Therefore, we have:
1/sin^2 9 = (5/4)^2
Simplifying, we get sin^2 9 = 16/25. Taking the square root on both sides gives sin 9 = 4/5.
Now, we can substitute the value of sin 9 into the identity cot^2 9 + 1 = csc^2 9:
cot^2 9 + 1 = (1/sin^2 9)
Substituting sin 9 = 4/5, we get:
cot^2 9 + 1 = (1/(4/5)^2)
Simplifying, we find that cot 9 = 3/4.