Simplify sec(x)csc(x)

Problem

sec(x)*csc(x)

Solution

1. Rewrite the trigonometric functions in terms of sine and cosine using the reciprocal identities sec(x)=1/cos(x) and csc(x)=1/sin(x)

sec(x)*csc(x)=1/cos(x)⋅1/sin(x)

2. Multiply the fractions to combine the expression into a single term.

1/cos(x)⋅1/sin(x)=1/(sin(x)*cos(x))

3. Apply the double angle identity for sine, sin(2*x)=2*sin(x)*cos(x) which implies sin(x)*cos(x)=sin(2*x)/2

1/(sin(x)*cos(x))=1/sin(2*x)/2

4. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

1/sin(2*x)/2=2/sin(2*x)

5. Use the reciprocal identity 1/sin(θ)=csc(θ) to write the final simplified form.

2/sin(2*x)=2*csc(2*x)

Final Answer

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

Want more problems? Check here!