What is the general solution for sin(x)*tan(x)+1=sin(x)+tan(x) ?

The general solution for sin(x)*tan(x)+1=sin(x)+tan(x) is x= pi/4+pin

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sin(x)*tan(x)+1=sin(x)+tan(x)

sin (x) \cdot tan (x) + 1 = sin (x) + tan (x)

x = \frac{π}{4} + πn

Solution steps

sin (x) tan (x) + 1 = sin (x) + tan (x)

Subtract

sin (x) + tan (x)

from both sides

sin (x) tan (x) + 1 - sin (x) - tan (x) = 0

Express with sin, cos

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

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

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

Factor

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

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

Solving each part separately

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

1 - sin (x) = 0 : x = \frac{π}{2} + 2 πn

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

Combine all the solutions

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

Since the equation is undefined for:

\frac{π}{2} + 2 πn

x = \frac{π}{4} + πn

1.33 \cdot sin (5 6^{\circ}) = 1 \cdot sin (x)

cosh (x) = sinh (x)

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

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

\frac{- cos ( x ) + sin ( x )}{1 + sin ( 2 x )} = 0

What is the general solution for sin(x)*tan(x)+1=sin(x)+tan(x) ?
The general solution for sin(x)*tan(x)+1=sin(x)+tan(x) is x= pi/4+pin