Solve for x sec(x)^2(1-sin(x)^2)=1

Problem

sec2(x)*(1−sin2(x))=1

Solution

1. Identify the trigonometric identity for the expression inside the parentheses.

2. Apply the Pythagorean identity sin2(x)+cos2(x)=1 which implies 1−sin2(x)=cos2(x)

3. Substitute the identity into the original equation.

sec2(x)*cos2(x)=1

4. Use the reciprocal identity sec(x)=1/cos(x) which means sec2(x)=1/cos2(x)

5. Simplify the left side of the equation by multiplying the terms.

1/cos2(x)⋅cos2(x)=1

1=1

6. Conclude that since the equation simplifies to a true statement for all values where the expression is defined, the equation is an identity. The equation holds true for all x in the domain of sec(x)

7. Determine the domain constraints for sec(x) which is undefined where cos(x)=0

x≠π/2+n*π

Final Answer

sec2(x)*(1−sin2(x))=1⇒1=1

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