What is the general solution for tan(x+15)*cot(2x-5)=1 ?

The general solution for tan(x+15)*cot(2x-5)=1 is x=-2pin+20,x=-pi+20-2pin

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tan(x+15)*cot(2x-5)=1

tan (x + 15) \cdot cot (2 x - 5) = 1

x = - 2 πn + 20, x = - π + 20 - 2 πn

Solution steps

tan (x + 15) cot (2 x - 5) = 1

Subtract

1

from both sides

tan (x + 15) cot (2 x - 5) - 1 = 0

Express with sin, cos

\frac{cos ( - 5 + 2 x ) sin ( 15 + x ) - cos ( 15 + x ) sin ( - 5 + 2 x )}{cos ( 15 + x ) sin ( - 5 + 2 x )} = 0

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

cos (- 5 + 2 x) sin (15 + x) - cos (15 + x) sin (- 5 + 2 x) = 0

Rewrite using trig identities

sin (15 + x - (- 5 + 2 x)) = 0

General solutions for

sin (15 + x - (- 5 + 2 x)) = 0

15 + x - (- 5 + 2 x) = 0 + 2 πn, 15 + x - (- 5 + 2 x) = π + 2 πn

Solve

15 + x - (- 5 + 2 x) = 0 + 2 πn : x = - 2 πn + 20

Solve

15 + x - (- 5 + 2 x) = π + 2 πn : x = - π + 20 - 2 πn

x = - 2 πn + 20, x = - π + 20 - 2 πn

cos^{2} (x) + 2 cos (x) + 1 = 0, 0 \leq x \leq 2 π

x, sin (2 x) = \frac{1}{2}

- 4 sin (x) = - cos^{2} (x) + 4, 0 \leq x \leq 2 π

2 sin (θ) cos (θ) - cos (θ) = 0

sin (2 x) cot (x) = 0

What is the general solution for tan(x+15)*cot(2x-5)=1 ?
The general solution for tan(x+15)*cot(2x-5)=1 is x=-2pin+20,x=-pi+20-2pin