Solve for x tan(x)^2+sec(x)-1=0

Problem

tan(x)+sec(x)−1=0

Solution

1. Apply the Pythagorean identity to rewrite the equation in terms of a single trigonometric function. Use tan(x)=sec(x)−1

sec(x)−1+sec(x)−1=0

2. Combine like terms to form a quadratic equation in terms of sec(x)

sec(x)+sec(x)−2=0

3. Factor the quadratic expression by looking for two numbers that multiply to −2 and add to 1

(sec(x)+2)*(sec(x)−1)=0

4. Set each factor to zero to find the possible values for sec(x)

sec(x)=−2

sec(x)=1

5. Convert to cosine using the identity cos(x)=1/sec(x)

cos(x)=−1/2

cos(x)=1

6. Solve for x within the general solution set. For cos(x)=−1/2 x=(2*π)/3+2*n*π or x=(4*π)/3+2*n*π For cos(x)=1 x=2*n*π

x=(2*π)/3+2*n*π

x=(4*π)/3+2*n*π

x=2*n*π

Final Answer

tan(x)+sec(x)−1=0⇒x∈{2*n*π,(2*π)/3+2*n*π,(4*π)/3+2*n*π}

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