Verify the Identity (cos(x))/(sec(x))+(sin(x))/(csc(x))=1

Problem

cos(x)/sec(x)+sin(x)/csc(x)=1

Solution

1. Identify the reciprocal identities for the functions in the denominators.

sec(x)=1/cos(x)

csc(x)=1/sin(x)

2. Substitute these reciprocal identities into the left side of the equation.

cos(x)/1/cos(x)+sin(x)/1/sin(x)

3. Simplify the complex fractions by multiplying the numerators by the reciprocals of the denominators.

cos(x)⋅cos(x)+sin(x)⋅sin(x)

cos2(x)+sin2(x)

4. Apply the Pythagorean identity, which states that the sum of the squares of sine and cosine is equal to 1.

cos2(x)+sin2(x)=1

Final Answer

cos(x)/sec(x)+sin(x)/csc(x)=1

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