What is the general solution for tan(θ/2)-sin(θ)=0 ?

The general solution for tan(θ/2)-sin(θ)=0 is θ= pi/2+2pin,θ=(3pi)/2+2pin,θ=4pin,θ=2pi+4pin

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tan(θ/2)-sin(θ)=0

tan (\frac{θ}{2}) - sin (θ) = 0

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = 4 πn, θ = 2 π + 4 πn

Solution steps

tan (\frac{θ}{2}) - sin (θ) = 0

Express with sin, cos

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

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

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

Rewrite using trig identities

- cos (θ) sin (\frac{θ}{2}) = 0

Solving each part separately

cos (θ) = 0 or sin (\frac{θ}{2}) = 0

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

sin (\frac{θ}{2}) = 0 : θ = 4 πn, θ = 2 π + 4 πn

Combine all the solutions

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = 4 πn, θ = 2 π + 4 πn

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

cos (θ) cos (2 θ) + sin (θ) sin (2 θ) = \frac{2}{2}

sec^{2} (x) - 2 tan (x) = 4

cos (4 x) = cos (2 x)

sin (4 x) + sin (2 x) = 0

What is the general solution for tan(θ/2)-sin(θ)=0 ?
The general solution for tan(θ/2)-sin(θ)=0 is θ= pi/2+2pin,θ=(3pi)/2+2pin,θ=4pin,θ=2pi+4pin