Question

if cosec theta - cot theta =2/3 then find the following ​

Answer 1

Answer:

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Answer 2

$csc^2(theta)$ is greater than 1, we know that there is no real solution for theta that satisfies the given equation. Therefore, we cannot find the value of -1.

We are given that $cosec(theta) - cot(theta) = 2/3.$

To find the value of -1, we can use the trigonometric identity:

$sin^2(theta)+ cos^2(theta) = 1$

Dividing both sides by $sin^2(theta)$, we get:

$1 + cot^2(theta) = csc^2(theta)$

Using the given equation, we can substitute $cosec(theta) = cot(theta) + 2/3:$

$1 + cot^2(theta) = (cot(theta) + 2/3)^2$

Expanding the right side and simplifying, we get:

$cot^2(theta) - 4/3 cot(theta) - 5/9 = 0$

We can solve this quadratic equation for cot(theta) using the quadratic formula:

$cot(theta) = [4/3 ± sqrt((4/3)^2 + 4*5/9)]/2\\cot(theta) = [2/3 ± sqrt(61)/3]$

Therefore, the possible values of cot(theta) are:

$cot(theta) = [2/3 + sqrt(61)/3] or cot(theta) = [2/3 - sqrt(61)/3]$

To find the value of -1, we can use the identity:

$cot^2(theta) + 1 = csc^2(theta)$

Substituting the two possible values of cot(theta), we get:

$csc^2(theta) = [2/3 + sqrt(61)/3]^2 + 1 = 1.214$

or

$csc^2(theta) = [2/3 - sqrt(61)/3]^2 + 1 = 11.956$

for such more question on theta

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