Solve for x cos(x)^2(1+tan(x)^2)=1

Problem

cos(x)*(1+tan(x))=1

Solution

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

2. Apply the identity 1+tan(x)=sec(x) to rewrite the expression.

cos(x)*sec(x)=1

3. Use the reciprocal identity sec(x)=1/cos(x) to substitute for the secant term.

cos(x)⋅1/cos(x)=1

4. Simplify the left side of the equation by canceling the common factors.

1=1

5. Conclude that since the equation simplifies to a true statement for all values where the expression is defined, the equation is an identity. The variable x can be any value except where tan(x) is undefined.

Final Answer

cos(x)*(1+tan(x))=1

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