Certainly! To prove the equation (cos(x) + sin(x))/(cos(x) - sin(x)) = sec(2x) + tan(2x), we need to use trigonometric identities and algebraic manipulations.
Step 1: Start with the left-hand side (LHS).
(cos(x) + sin(x))/(cos(x) - sin(x))
Step 2: To simplify this expression, we can multiply both the numerator and denominator by the conjugate of the denominator, which is (cos(x) + sin(x)).
[(cos(x) + sin(x)) * (cos(x) + sin(x))]/[(cos(x) - sin(x)) * (cos(x) + sin(x))]
Step 3: Expand the numerator and denominator.
[cos²(x) + 2cos(x)sin(x) + sin²(x)]/[cos²(x) - sin²(x)]
Step 4: Simplify the numerator.
[1 + 2cos(x)sin(x)]/[cos²(x) - sin²(x)]
Step 5: Recall the trigonometric identity: sin(2x) = 2sin(x)cos(x) and cos(2x) = cos²(x) - sin²(x).
[1 + sin(2x)]/[cos(2x)]
Step 6: Recall the trigonometric identity: sec(x) = 1/cos(x) and tan(x) = sin(x)/cos(x).
[sec(2x) + sin(2x) * tan(2x)]/[cos(2x)]
Step 7: Distribute tan(2x) term.
[sec(2x) + tan(2x) * sin(2x)]/[cos(2x)]
Step 8: Recognize that sin(2x)/cos(2x) is equal to tan(2x).
[sec(2x) + tan(2x)]/[cos(2x)]
Step 9: Since the numerator and denominator are the same as the right-hand side (RHS), we have proven that
(cos(x) + sin(x))/(cos(x) - sin(x)) = sec(2x) + tan(2x).
This completes the explanation and proof of the given equation.