tan x * tan x - tan x = 0
a. 0 + n*pi * pi/4 + 2n * pi
b. 0+r2 pi 5 pi 4 +2n pi
c. pi/2 + n*pi, (5pi)/4 + 2n * pi
d. pi/2 + ri * pi * pi/4 + 2n * pi
To solve the given equation: tan(x) * tan(x) - tan(x) = 0, we can factor out the common term tan(x):
tan(x) * (tan(x) - 1) = 0
Now, we have two factors, tan(x) and (tan(x) - 1), and the product of these two factors is equal to zero. According to the zero-product property, for a product to be zero, at least one of the factors must be zero.
So we can set each factor equal to zero and solve for x:
tan(x) = 0
Taking the inverse tangent (arctan) of both sides, we get:
x = arctan(0) = 0 + n * π, where n is an integer
Next, we solve for the second factor:
tan(x) - 1 = 0
tan(x) = 1
Taking the inverse tangent (arctan) of both sides, we get:
x = arctan(1) = π/4 + n * π, where n is an integer
Combining both solutions, we have:
x = 0 + n * π or x = π/4 + n * π, where n is an integer
Now let's compare the options to see which one matches our solution:
a. 0 + n * π, π/4 + 2n * π (This matches)
b. 0 + √2π, 5π/4 + 2n * π (This does not match)
c. π/2 + n * π, 5π/4 + 2n * π (This does not match)
d. π/2 + √i * π * π/4, 2n * π (This does not match)
Therefore, the correct answer is option a) 0 + n * π, π/4 + 2n * π.