Prove that cos 2x - 1 cos 2x + 1 = cosec x + cotx Show that cos 2x - 1 cos 2x + 1 = -tan x If tan = c, derive the expressions for cos A and sin A in terms of c. Hence, or otherwise, prove that if sec A - tan A = x Then, tan = x

Step 1: Start with the left-hand side (LHS) of the equation. LHS = sqrt((1+ x)/(1- x)) Step 2: Multiply the numerator and denominator inside the square root by the conjugate of the denominator, which is (1+ x). LHS = sqrt((1+ x)/(1- x) × (1+ x)/(1+ x)) LHS = sqrt(((1+ x)^2)/(1^2 - ^2 x)) Step 3: Apply the Pythagorean identity ^2 x + ^2 x = 1, which implies 1 - ^2 x = ^2 x. LHS = sqrt(((1+ x)^2)/(^2 x)) Step 4: Take the square root of the numerator and the denominator. Assuming 1+ x 0 and x > 0 for the principal value, we can remove the absolute values. LHS = (1+ x)/( x) Step 5: Separate the fraction into two terms. LHS = (1)/( x) + ( x)/( x) Step 6: Use the definitions of cosecant ( x = (1)/( x)) and cotangent ( x = ( x)/( x)). LHS = x + x This matches the right-hand side (RHS) of the equation. Therefore, we have proven that sqrt((1+ x)/(1- x)) = x + x. Send me the next one 📸