What is the general solution for tan(45+x)+tan(45-x)=4 ?

The general solution for tan(45+x)+tan(45-x)=4 is x=150+180n,x=30+180n

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tan(45+x)+tan(45-x)=4

tan (4 5^{\circ} + x) + tan (4 5^{\circ} - x) = 4

x = 15 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

Solution steps

tan (4 5^{\circ} + x) + tan (4 5^{\circ} - x) = 4

Rewrite using trig identities

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

Subtract

4

from both sides

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

Simplify

\frac{2 cos ^{2} ( x ) + 2 sin ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} - 4 : \frac{- 2 cos ^{2} ( x ) + 6 sin ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )}

\frac{- 2 cos ^{2} ( x ) + 6 sin ^{2} ( x )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} = 0

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

- 2 cos^{2} (x) + 6 sin^{2} (x) = 0

Factor

- 2 cos^{2} (x) + 6 sin^{2} (x) : 2 (3 sin (x) + cos (x)) (3 sin (x) - cos (x))

2 (3 sin (x) + cos (x)) (3 sin (x) - cos (x)) = 0

Solving each part separately

3 sin (x) + cos (x) = 0 or 3 sin (x) - cos (x) = 0

3 sin (x) + cos (x) = 0 : x = 15 0^{\circ} + 18 0^{\circ} n

3 sin (x) - cos (x) = 0 : x = 3 0^{\circ} + 18 0^{\circ} n

Combine all the solutions

x = 15 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

1 - cos (θ) = sin (θ)

3 cot (\frac{3 π}{7} x) + 3 = 0

sin (u) = 1

tan (x) - 1 = \frac{1}{3} - \frac{1}{3} cot (x)

22 cos (x) + 3 = 5

What is the general solution for tan(45+x)+tan(45-x)=4 ?
The general solution for tan(45+x)+tan(45-x)=4 is x=150+180n,x=30+180n