What is the general solution for (cot(θ)+csc(θ))/(sec(θ)+1)=sin(θ) ?

The general solution for (cot(θ)+csc(θ))/(sec(θ)+1)=sin(θ) is θ=0.90455…+2pin,θ=2pi-0.90455…+2pin

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(cot(θ)+csc(θ))/(sec(θ)+1)=sin(θ)

\frac{cot ( θ ) + csc ( θ )}{sec ( θ ) + 1} = sin (θ)

θ = 0.90455 \dots + 2 πn, θ = 2 π - 0.90455 \dots + 2 πn

Solution steps

\frac{cot ( θ ) + csc ( θ )}{sec ( θ ) + 1} = sin (θ)

Subtract

sin (θ)

from both sides

\frac{cot ( θ ) + csc ( θ )}{sec ( θ ) + 1} - sin (θ) = 0

Simplify

\frac{cot ( θ ) + csc ( θ )}{sec ( θ ) + 1} - sin (θ) : \frac{cot ( θ ) + csc ( θ ) - sin ( θ ) ( sec ( θ ) + 1 )}{sec ( θ ) + 1}

\frac{cot ( θ ) + csc ( θ ) - sin ( θ ) ( sec ( θ ) + 1 )}{sec ( θ ) + 1} = 0

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

cot (θ) + csc (θ) - sin (θ) (sec (θ) + 1) = 0

Express with sin, cos

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

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

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

Factor

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

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

Solving each part separately

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

1 + cos (θ) = 0 : θ = π + 2 πn

cos (θ) - sin^{2} (θ) = 0 : θ = arccos (\frac{- 1 + 5}{2}) + 2 πn, θ = 2 π - arccos (\frac{- 1 + 5}{2}) + 2 πn

Combine all the solutions

θ = π + 2 πn, θ = arccos (\frac{- 1 + 5}{2}) + 2 πn, θ = 2 π - arccos (\frac{- 1 + 5}{2}) + 2 πn

Since the equation is undefined for:

π + 2 πn

θ = arccos (\frac{- 1 + 5}{2}) + 2 πn, θ = 2 π - arccos (\frac{- 1 + 5}{2}) + 2 πn

Show solutions in decimal form

θ = 0.90455 \dots + 2 πn, θ = 2 π - 0.90455 \dots + 2 πn

2 = cos^{2} (x)

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

\frac{1}{cos ( 2 x )} + tan (2 x) = 3 cos (2 x), 0^{\circ} < x < 9 0^{\circ}

sin (x) = \frac{4}{5}, 0 \leq x < 2 π

7 sin^{2} (θ) - 5 sin (θ) = 2

What is the general solution for (cot(θ)+csc(θ))/(sec(θ)+1)=sin(θ) ?
The general solution for (cot(θ)+csc(θ))/(sec(θ)+1)=sin(θ) is θ=0.90455…+2pin,θ=2pi-0.90455…+2pin