What is the general solution for tan(x+pi/4)-tan(x-pi/4)=3 ?

The general solution for tan(x+pi/4)-tan(x-pi/4)=3 is x=-0.42053…+pin,x=0.42053…+pin

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tan(x+pi/4)-tan(x-pi/4)=3

tan (x + \frac{π}{4}) - tan (x - \frac{π}{4}) = 3

x = - 0.42053 \dots + πn, x = 0.42053 \dots + πn

Solution steps

tan (x + \frac{π}{4}) - tan (x - \frac{π}{4}) = 3

Rewrite using trig identities

\frac{2 sin ^{2} ( x ) + 2 cos ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} = 3

Subtract

3

from both sides

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

Simplify

\frac{2 sin ^{2} ( x ) + 2 cos ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} - 3 : \frac{5 sin ^{2} ( x ) - cos ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}

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

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

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

Factor

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

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

Solving each part separately

5 sin (x) + cos (x) = 0 or 5 sin (x) - cos (x) = 0

5 sin (x) + cos (x) = 0 : x = arctan (- \frac{5}{5}) + πn

5 sin (x) - cos (x) = 0 : x = arctan (\frac{5}{5}) + πn

Combine all the solutions

x = arctan (- \frac{5}{5}) + πn, x = arctan (\frac{5}{5}) + πn

Show solutions in decimal form

x = - 0.42053 \dots + πn, x = 0.42053 \dots + πn

cos (x) (2 sin (x) - 2) = 0

2 sin^{2} (θ) - 3 sin (θ) = - 1

4 cos^{2} (θ) + 2 = 3

cos (kπ) = 0

arcsin (x + 1) = - 0.5

What is the general solution for tan(x+pi/4)-tan(x-pi/4)=3 ?
The general solution for tan(x+pi/4)-tan(x-pi/4)=3 is x=-0.42053…+pin,x=0.42053…+pin