Solve ; 1 + (cotx)^2 = 2 (tanx)^2

csc²(x) = 1 + cot²(x), so your equation is

csc²(x) = 2tan²(x) ==> csc(x) = ±√(2) tan(x) ==> (use csc(x) =1/sin(x) and tan(x) = sin(x)/cos(x))

cos(x) = ±√(2) sin²(x) = ±√(2)(1 - cos²(x)) ==>

(√(2)cos(x) - 1)(cos(x) + √(2)) = 0 or

(√(2)cos(x) + 1)(cos(x) - √(2)) = 0.

The second factor in each of these has no root since |cos(x)| ≤ 1 and √(2) > 1.

So all answers are given by cos(x) = ±1/√(2). The general solution is

x = π/4 + nπ/2, for any integer n.

On the interval [0, 2π), the four solutions are {π/4, 3π/4, 5π/4, 7π/4}.

(Just and FYI: You can actually guess the solution set from csc²(x) = 2tan²(x). But I worked it out to make sure I didn't miss any solutions.)

Solve ; 1 + (cotx)^2 = 2 (tanx)^2

1 + 1 / (tanx)^2 = 2 (tanx)^2

(tanx)^2 + 1 = 2(tanx)^4

2(tanx)^4 -(tanx)^2 -1 = 0

[2(tanx)^2 + 1][(tanx)^2 - 1] = 0

2(tanx)^2 =1 = 0, ===> tanx = +/-sqrt(1/2)

x = 35.3, x = 144.7, x = 215.3, and -35.3 >=======< ANSWER
. . . . . . D E G R E E S.
ALSO, tanx = +/-(1)

x = 45, x=135, x=225, and x = -45 >=============< ANSWER