Answer:
tan(x) = 1
Step-by-step explanation:
I find the least tedious way to answer questions like this is to use a graphing calculator to graph a function that is zero for the desired x. We can write that function by subtracting csc(x) from both sides of the equation:
0 = sin(x)tan(x) +cos(x) -csc(x)
The graph shows a solution to be x=π/4, corresponding to 45°. The values of the trig functions for this angle are ...
- sin(π/4) = (√2)/2
- cos(π/4) = (√2)/2
- tan(π/4) = 1
- csc(π/4) = √2
_____
An algebraic solution might go like this.
1/sin(x) = sin(x)·tan(x) +cos(x)
Multiplying by sin(x), we get ...
1 = tan(x)·sin(x)^2 + sin(x)cos(x)
Substituting for sin(x)^2, this is
1 = tan(x)(1 -cos(x)^2) +sin(x)cos(x)
1 = tan(x) - (sin(x)/cos(x))·cos(x)^2 +sin(x)cos(x) . . . using tan(x) = sin(x)/cos(x)
1 = tan(x)
