Solve the equation by factoring. (enter your answers as a comma-separated list. let k be any integer. round terms to three decimal places where appropriate. if there is no solution, enter no solution.) csc (θ) cot (θ) - sin (θ) tan (θ) = cos (θ)

Answer: There is no solution

Step-by-step explanation:

Given that

csc (θ) cot (θ) - sin (θ) tan (θ) = cos (θ) ... (1)

Where

csc (θ) = 1/sin (θ)

cot (θ) = 1/tan (θ) = cos (θ) / sin (θ)

tan (θ) = sin (θ) / cos (θ)

Substitute them in equation (1)

1/sin (θ) * cos (θ) / sin (θ) - sin (θ) * sin (θ) / cos (θ) = cos (θ)

Cos (θ) / sin^2 (θ) - sin^2 (θ) / cos (θ) = cos (θ)

Cos (θ) / sin^2 (θ) = sin^2 (θ) / cos (θ) + cos (θ)

Take the LCM of right hand side

Cos (θ) / sin^2 (θ) = (sin^2 (θ) + cos^2 (θ)) / cos (θ)

But sin^2 (θ) + cos^2 (θ) = 1

Cos (θ) / sin^2 (θ) = 1 / cos (θ)

Cross multiply

Cos^2 (θ) = sin^2 (θ)

Cos^2 (θ) - sin^2 (θ) = 0

If θ = 45

That means there's no solution