What is the general solution for 10tan(θ)sec(θ)=10cot(θ)csc(θ) ?

The general solution for 10tan(θ)sec(θ)=10cot(θ)csc(θ) is θ= pi/4+pin

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10tan(θ)sec(θ)=10cot(θ)csc(θ)

10 tan (θ) sec (θ) = 10 cot (θ) csc (θ)

θ = \frac{π}{4} + πn

Solution steps

10 tan (θ) sec (θ) = 10 cot (θ) csc (θ)

Subtract

10 cot (θ) csc (θ)

from both sides

10 tan (θ) sec (θ) - 10 cot (θ) csc (θ) = 0

Express with sin, cos

\frac{- 10 cos ^{3} ( θ ) + 10 sin ^{3} ( θ )}{cos ^{2} ( θ ) sin ^{2} ( θ )} = 0

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

- 10 cos^{3} (θ) + 10 sin^{3} (θ) = 0

Factor

- 10 cos^{3} (θ) + 10 sin^{3} (θ) : 10 (sin (θ) - cos (θ)) (sin^{2} (θ) + sin (θ) cos (θ) + cos^{2} (θ))

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

Rewrite using trig identities

10 (- cos (θ) + sin (θ)) (cos (θ) sin (θ) + 1) = 0

Solving each part separately

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

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

cos (θ) sin (θ) + 1 = 0 :

No Solution

Combine all the solutions

θ = \frac{π}{4} + πn

arctan (x) = \frac{( 2 \cdot 1.8 )}{[ ( 3170 + 1.7 ) - 1 ]}

cos (4 x) + sin (\frac{x}{2}) = 0

2 sin^{2} (θ) - 5 sin (θ) = 3

cos (3 x - \frac{π}{4}) = \frac{1}{2}, 0 \leq x \leq π

4 sin^{2} (x) = 3, 0 \leq x \leq 10 8^{\circ}

What is the general solution for 10tan(θ)sec(θ)=10cot(θ)csc(θ) ?
The general solution for 10tan(θ)sec(θ)=10cot(θ)csc(θ) is θ= pi/4+pin