How do you find the exact value of $cot^2theta=csctheta+1$ in the interval $0<=theta<360$ ?

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Answer 1 · 2016-11-17T21:48:18

Please see the explanation.

Substitute $cos^2(theta)/sin^2(theta)$ for $cot^2(theta)$ and $1/sin(theta)$ for $csc(theta)$ :

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

Multiply both sides by $sin^2(theta)$ :

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

Substitute $1 - sin^2(theta)$ for $cos^2(theta)$

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

Combine like terms:

$2sin^2(theta) + sin(theta) - 1 = 0$

Factor

$(2sin(theta) - 1)(sin(theta) + 1) = 0$

Set each factor equal to zero:

$(2sin(theta) - 1) = 0 and (sin(theta) + 1) = 0$

Solve both equations for $sin(theta)$ ;

$sin(theta) = 1/2 and sin(theta) = -1$

The left equation happens twice; the right one only once:

$theta = sin^-1(1/2), theta = 180 - sin^-1(1/2), and theta = sin^-1(-1)$

$theta = 30^@, theta = 150^@, and theta = 270^@$

How do you find the exact value of cot^2theta=csctheta+1cot^2theta=csctheta+1 in the interval 0<=theta<3600<=theta<360?