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

Problem

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

Solution

1. Apply reciprocal identities to rewrite the denominators in terms of sine and cosine.

csc(x)=1/sin(x)

sec(x)=1/cos(x)

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

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

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

sin(x)⋅sin(x)+cos(x)⋅cos(x)

sin2(x)+cos2(x)

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

sin2(x)+cos2(x)=1

Final Answer

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

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