Verify the Identity sec(pi/2-x)=csc(x)

Problem

sec(π/2−x)=csc(x)

Solution

1. Use the reciprocal identity for the secant function to rewrite the expression in terms of cosine.

sec(π/2−x)=1/cos(π/2−x)

2. Apply the cosine difference formula, which states cos(A−B)=cos(A)*cos(B)+sin(A)*sin(B) where A=π/2 and B=x

cos(π/2−x)=cos(π/2)*cos(x)+sin(π/2)*sin(x)

3. Substitute the known values for cos(π/2)=0 and sin(π/2)=1 into the expression.

cos(π/2−x)=(0)*cos(x)+(1)*sin(x)

4. Simplify the denominator to find the equivalent sine expression.

cos(π/2−x)=sin(x)

5. Substitute the result back into the reciprocal identity.

1/cos(π/2−x)=1/sin(x)

6. Apply the reciprocal identity for cosecant, where 1/sin(x)=csc(x)

1/sin(x)=csc(x)

Final Answer

sec(π/2−x)=csc(x)

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