secx^2 - secx=2

Question

Find all exact solutions no decimals

Answer 1

cos²(x)(sec²(x)-sec(x))=2cos²(x),

1-cos(x)=2cos²(x),

2cos²(x)+cos(x)-1=0 is a quadratic equation.

2cos²(x)+cos(x)-1=(2cos(x)-1)(cos(x)+1)=0.

Therefore cos(x)=½ or cos(x)=-1.

x=π/3+2πn or 5π/3+2πn, where n is an integer (positive or negative); or:

x=(2n+1)π (odd multiples of π, including negative multiples).