What is the general solution for (cot(x))/(sec(x))=sin(x) ?

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

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(cot(x))/(sec(x))=sin(x)

\frac{cot ( x )}{sec ( x )} = sin (x)

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

Solution steps

\frac{cot ( x )}{sec ( x )} = sin (x)

Subtract

sin (x)

from both sides

\frac{cot ( x )}{sec ( x )} - sin (x) = 0

Simplify

\frac{cot ( x )}{sec ( x )} - sin (x) : \frac{cot ( x ) - sin ( x ) sec ( x )}{sec ( x )}

\frac{cot ( x ) - sin ( x ) sec ( x )}{sec ( x )} = 0

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

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

Express with sin, cos

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

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

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

Rewrite using trig identities

cos (2 x) = 0

General solutions for

cos (2 x) = 0

2 x = \frac{π}{2} + 2 πn, 2 x = \frac{3 π}{2} + 2 πn

Solve

2 x = \frac{π}{2} + 2 πn : x = \frac{π}{4} + πn

Solve

2 x = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{4} + πn

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

tan^{2} (x) - sec^{2} (x) = cos (- x)

sec^{2} (θ) tan (θ) = 2 tan (θ), 0 \leq θ \leq 2 π

cos (2 x) cos (x) - sin (2 x) sin (x) = \frac{2}{2}

- 2 + cos (θ) = - 2 + \frac{6}{5} \cdot cos (θ)

tan^{2} (y) + 3 - 2 sec^{2} (y) = 0

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