What is the general solution for sin(x)+cos(x)+cot(x)=csc(x) ?

The general solution for sin(x)+cos(x)+cot(x)=csc(x) is x= pi/2+2pin,x=(3pi)/2+2pin

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sin(x)+cos(x)+cot(x)=csc(x)

sin (x) + cos (x) + cot (x) = csc (x)

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

Solution steps

sin (x) + cos (x) + cot (x) = csc (x)

Subtract

csc (x)

from both sides

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

Express with sin, cos

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

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

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

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

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

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

Combine all the solutions

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

Since the equation is undefined for:

2 πn + 2 π

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

Merge Overlapping Intervals

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

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

cos (x) = \frac{3}{13}

3 tan^{3} (x) - 9 tan (x) = 0

cot (θ) = - 1, cos (θ), 9 0^{\circ} < θ < 18 0^{\circ}

2 sin (3 θ) + 1 = 0, 0 \leq θ \leq 2 π

What is the general solution for sin(x)+cos(x)+cot(x)=csc(x) ?
The general solution for sin(x)+cos(x)+cot(x)=csc(x) is x= pi/2+2pin,x=(3pi)/2+2pin