Find a numerical value of one trigonometric function of x for Cscx = sin xtan x + COS X.

  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 ...

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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)