csc(x)-sin(x)=cos(x)cot(3x-50)

Solution

csc (x) - sin (x) = cos (x) cot (3 x - 50)

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = πn + 25, x = \frac{π}{2} + 25 + πn

Solution steps

csc (x) - sin (x) = cos (x) cot (3 x - 50)

Subtract

cos (x) cot (3 x - 50)

from both sides

csc (x) - sin (x) - cos (x) cot (3 x - 50) = 0

Express with sin, cos

\frac{sin ( - 50 + 3 x ) - sin ( - 50 + 3 x ) sin ^{2} ( x ) - cos ( - 50 + 3 x ) cos ( x ) sin ( x )}{sin ( - 50 + 3 x ) sin ( x )} = 0

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

sin (- 50 + 3 x) - sin (- 50 + 3 x) sin^{2} (x) - cos (- 50 + 3 x) cos (x) sin (x) = 0

Rewrite using trig identities

cos^{2} (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) cos (x) sin (x) = 0

Factor

cos^{2} (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) cos (x) sin (x) : cos (x) (cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x))

cos (x) (cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x)) = 0

Solving each part separately

cos (x) = 0 or cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) sin (- 50 + 3 x) - cos (- 50 + 3 x) sin (x) = 0 : x = πn + 25, x = \frac{π}{2} + 25 + πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = πn + 25, x = \frac{π}{2} + 25 + πn

3 cot (θ) = - 3

tan (2 x) = \frac{2}{3}

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

sin (θ) = \frac{5}{6}, cos (θ)

tan^{2} (t) - sec^{2} (t) = 1