What is the general solution for sec(x)tan(x)-cos(x)cot(x)=sin(x) ?

The general solution for sec(x)tan(x)-cos(x)cot(x)=sin(x) is x= pi/4+pin,x=(3pi)/4+pin

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sec(x)tan(x)-cos(x)cot(x)=sin(x)

sec (x) tan (x) - cos (x) cot (x) = sin (x)

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

Solution steps

sec (x) tan (x) - cos (x) cot (x) = sin (x)

Subtract

sin (x)

from both sides

sec (x) tan (x) - cos (x) cot (x) - sin (x) = 0

Express with sin, cos

\frac{- cos ^{4} ( x ) + sin ^{2} ( x ) - cos ^{2} ( x ) sin ^{2} ( x )}{cos ^{2} ( x ) sin ( x )} = 0

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

- cos^{4} (x) + sin^{2} (x) - cos^{2} (x) sin^{2} (x) = 0

Rewrite using trig identities

- cos^{4} (x) + sin^{4} (x) = 0

Factor

- cos^{4} (x) + sin^{4} (x) : (sin^{2} (x) + cos^{2} (x)) (sin (x) + cos (x)) (sin (x) - cos (x))

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

Rewrite using trig identities

(- cos (x) + sin (x)) (cos (x) + sin (x)) = 0

Solving each part separately

- cos (x) + sin (x) = 0 or cos (x) + sin (x) = 0

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

cos (x) + sin (x) = 0 : x = \frac{3 π}{4} + πn

Combine all the solutions

x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

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

sin (2 x) = - 5 cos (2 x)

tan (x) cot (x) = sec (x) csc (x)

4 sin (x) = cos (x) - 2

sin (θ) = 1 - cos (θ)

What is the general solution for sec(x)tan(x)-cos(x)cot(x)=sin(x) ?
The general solution for sec(x)tan(x)-cos(x)cot(x)=sin(x) is x= pi/4+pin,x=(3pi)/4+pin