Solve for x sec(x)csc(x)=2csc(x)

Problem

sec(x)*csc(x)=2*csc(x)

Solution

1. Subtract 2*csc(x) from both sides to set the equation to zero.

sec(x)*csc(x)−2*csc(x)=0

2. Factor out the common term csc(x) from the expression.

csc(x)*(sec(x)−2)=0

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

csc(x)=0

sec(x)−2=0

4. Analyze the first factor csc(x)=0 Since csc(x)=1/sin(x) there are no values of x for which csc(x)=0 because a fraction with a constant numerator of 1 can never equal zero.

csc(x)≠0

5. Solve the second factor sec(x)−2=0 by isolating the secant function.

sec(x)=2

6. Rewrite the equation using the reciprocal identity sec(x)=1/cos(x)

1/cos(x)=2

cos(x)=1/2

7. Determine the values of x within one period [0,2*π) where the cosine is 1/2

x=π/3

x=(5*π)/3

8. Generalize the solution by adding multiples of the period 2*π*n where n is an integer.

x=π/3+2*π*n

x=(5*π)/3+2*π*n

Final Answer

sec(x)*csc(x)=2*csc(x)⇒x=π/3+2*π*n,(5*π)/3+2*π*n

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