Solve for x (tan(x)-1)(sec(x)-1)=0

Problem

(tan(x)−1)*(sec(x)−1)=0

Solution

1. Apply the zero product property by setting each factor equal to zero.

tan(x)−1=0

sec(x)−1=0

2. Isolate the trigonometric functions in each equation.

tan(x)=1

sec(x)=1

3. Solve for x in the first equation tan(x)=1 The tangent function is 1 at π/4 in the first quadrant and (5*π)/4 in the third quadrant.

x=π/4+n*π

4. Solve for x in the second equation sec(x)=1 Since sec(x)=1/cos(x) this is equivalent to cos(x)=1

x=2*n*π

5. Combine the solutions into a general form where n is any integer.

x=π/4+n*π,2*n*π

Final Answer

x=π/4+n*π,2*n*π

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