What is the general solution for arctan(x)+arctan(3x)+arctan(7x)=180 ?

The general solution for arctan(x)+arctan(3x)+arctan(7x)=180 is No Solution for x\in\mathbb{R}

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arctan(x)+arctan(3x)+arctan(7x)=180

arctan (x) + arctan (3 x) + arctan (7 x) = 18 0^{\circ}

No Solution for x \in R

Solution steps

arctan (x) + arctan (3 x) + arctan (7 x) = 18 0^{\circ}

Subtract

18 0^{\circ}

from both sides

arctan (x) + arctan (3 x) + arctan (7 x) - 18 0^{\circ} = 0

Rewrite using trig identities

- 18 0^{\circ} + arctan (\frac{\frac{10 x}{1 - 21 x ^{2}} + x}{1 - \frac{10 x}{1 - 21 x ^{2}} x}) = 0

Move

18 0^{\circ}

to the right side

arctan (\frac{\frac{10 x}{1 - 21 x ^{2}} + x}{1 - \frac{10 x}{1 - 21 x ^{2}} x}) = 18 0^{\circ}

- \frac{π}{2} < arctan (x) < \frac{π}{2}

No Solution

Verify solutions by plugging them into the original equation

No Solution for x \in R

2 cos (a) = 1

cos^{2} (x) = (\frac{1}{2})

3 sin (\frac{π}{2} x) = 2

3 cos (A) - 1 = 0

cot (\frac{x}{2}) = - 1

What is the general solution for arctan(x)+arctan(3x)+arctan(7x)=180 ?
The general solution for arctan(x)+arctan(3x)+arctan(7x)=180 is No Solution for x\in\mathbb{R}