What is the general solution for tan(θ)sec(θ)=cot(θ)csc(θ) ?

The general solution for tan(θ)sec(θ)=cot(θ)csc(θ) is θ= pi/4+pin

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tan(θ)sec(θ)=cot(θ)csc(θ)

tan (θ) sec (θ) = cot (θ) csc (θ)

θ = \frac{π}{4} + πn

Solution steps

tan (θ) sec (θ) = cot (θ) csc (θ)

Subtract

cot (θ) csc (θ)

from both sides

tan (θ) sec (θ) - cot (θ) csc (θ) = 0

Express with sin, cos

\frac{- cos ^{3} ( θ ) + sin ^{3} ( θ )}{cos ^{2} ( θ ) sin ^{2} ( θ )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- cos^{3} (θ) + sin^{3} (θ) = 0

Factor

- cos^{3} (θ) + sin^{3} (θ) : (sin (θ) - cos (θ)) (sin^{2} (θ) + sin (θ) cos (θ) + cos^{2} (θ))

(sin (θ) - cos (θ)) (sin^{2} (θ) + sin (θ) cos (θ) + cos^{2} (θ)) = 0

Rewrite using trig identities

(- cos (θ) + sin (θ)) (cos (θ) sin (θ) + 1) = 0

Solving each part separately

- cos (θ) + sin (θ) = 0 or cos (θ) sin (θ) + 1 = 0

- cos (θ) + sin (θ) = 0 : θ = \frac{π}{4} + πn

cos (θ) sin (θ) + 1 = 0 :

No Solution

Combine all the solutions

θ = \frac{π}{4} + πn

0 = 4 + 3 cos (x)

cos^{2} (x) - cos (x) = 0.75

2 sin^{3} (x) + cos^{2} (x) = 1

2 tan (x) = tan (x) tan (x)

\frac{2 tan ( x )}{1 - tan ^{2} ( x )} = 3

What is the general solution for tan(θ)sec(θ)=cot(θ)csc(θ) ?
The general solution for tan(θ)sec(θ)=cot(θ)csc(θ) is θ= pi/4+pin