Question

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 1

Answer: There is no solution

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