tanx=cotx
Both cot x and tan x must either be 1 or -1, since tan x = 1/cot x
tan^2 x = 1
tan x = +/-1
tan 1 = 1 at 45 and 225 degrees. tan x = -1 at 135 and 315 degrees.
The equation tan(x) = cot(x) implies that the tangent of an angle x is equal to the cotangent of the same angle x.
To solve this equation, we need to simplify the given trigonometric expressions.
Let's start by recalling the definitions:
- The tangent of an angle x is defined as the ratio of the sine of x to the cosine of x: tan(x) = sin(x) / cos(x).
- The cotangent of an angle x is defined as the reciprocal of the tangent of x: cot(x) = 1 / tan(x).
Substituting the definitions into the equation, we have:
sin(x) / cos(x) = 1 / (sin(x) / cos(x))
To simplify this equation, we can multiply both sides by cos(x) to eliminate the denominator:
sin(x) = 1 / sin(x)
Now we can solve for sin(x) by cross-multiplying:
sin(x) * sin(x) = 1
Taking the square root of both sides:
sin(x) = ±1
Since the sine function is defined between -1 and 1, we can conclude that sin(x) = 1 or sin(x) = -1.
Case 1: sin(x) = 1
If sin(x) = 1, then x must be equal to π/2 (or 90 degrees) or any angle that differs by an integer multiple of 2π.
Case 2: sin(x) = -1
If sin(x) = -1, then x must be equal to -π/2 (or -90 degrees) or any angle that differs by an integer multiple of 2π.
Therefore, the solutions to the equation tan(x) = cot(x) are x = π/2 + 2πn and x = -π/2 + 2πn, where n is an integer.