Verify the Identity sec(x)=1/(csc(x))

Problem

sec(x)=1/csc(x)

Solution

1. Identify the standard reciprocal definitions for the trigonometric functions involved.

2. Recall that the secant function is defined as the reciprocal of the cosine function.

sec(x)=1/cos(x)

3. Recall that the cosecant function is defined as the reciprocal of the sine function.

csc(x)=1/sin(x)

4. Substitute the definition of cosecant into the right side of the given equation.

1/csc(x)=1/1/sin(x)

5. Simplify the complex fraction by multiplying by the reciprocal.

1/1/sin(x)=sin(x)

6. Compare the simplified right side to the left side.

sec(x)≠sin(x)

7. Conclude that the statement is not a trigonometric identity because sec(x) is the reciprocal of cos(x) not csc(x) The correct identity involving these terms is sec(x)=1/cos(x) or csc(x)=1/sin(x)

Final Answer

sec(x)≠1/csc(x)

Want more problems? Check here!