solve
(sq.rt 2) (tanx)(cosx) = tanx
sqrt2 cos x = 1
cosx = 1/sqrt2
x = 45 degrees and 315 degrees
tan x = 0 is also a solution. That leads to two more possible values of x.
See if you can figure out what they are.
To solve the equation (sqrt(2)) (tan(x))(cos(x)) = tan(x), we need to isolate the variable x. Here's how we can do it step by step:
Step 1: Simplify the equation
(sqrt(2)) (tan(x))(cos(x)) = tan(x)
Step 2: Divide both sides by tan(x)
(sqrt(2)) (cos(x)) = 1
Step 3: Divide both sides by (sqrt(2))
cos(x) = 1 / (sqrt(2))
Step 4: Simplify the right side
cos(x) = (sqrt(2)) / 2
Step 5: Determine the values of x
Since cos(x) = (sqrt(2)) / 2, we can find the angle values for x by looking at the unit circle. The unit circle tells us the cosine values for various angles.
In this case, the cosine value (sqrt(2)) / 2 is equivalent to cos(π/4). Therefore, one solution is x = π/4.
However, we need to consider that the tangent function has a period of π. So, the other solutions can be obtained by adding multiples of π to the angle π/4.
In general, the solutions for x will be:
x = π/4 + kπ, where k is an integer
So, the equation has an infinite number of solutions, where x could be π/4, 5π/4, 9π/4, and so on.