What is the general solution for arctan(x+2)-arctan(x+1)= pi/4 ?

The general solution for arctan(x+2)-arctan(x+1)= pi/4 is x=-1,x=-2

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arctan(x+2)-arctan(x+1)= pi/4

arctan (x + 2) - arctan (x + 1) = \frac{π}{4}

x = - 1, x = - 2

Solution steps

arctan (x + 2) - arctan (x + 1) = \frac{π}{4}

Rewrite using trig identities

arctan (\frac{x + 2 - ( x + 1 )}{1 + ( x + 2 ) ( x + 1 )}) = \frac{π}{4}

Apply trig inverse properties

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

Solve

\frac{x + 2 - ( x + 1 )}{1 + ( x + 2 ) ( x + 1 )} = 1 : x = - 1, x = - 2

x = - 1, x = - 2

Verify solutions by plugging them into the original equation

x = - 1, x = - 2

3 cos (2 x) = 1

cos^{2} (x) = (1 + sin (x)) (1 - cos (x))

tan (x) = 1, - π < x \leq π

\frac{1}{2 cos ^{2} ( x - 1 )} = \frac{1 + tan ^{2} ( x )}{2 sec ^{2} ( x )}

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

What is the general solution for arctan(x+2)-arctan(x+1)= pi/4 ?
The general solution for arctan(x+2)-arctan(x+1)= pi/4 is x=-1,x=-2