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(sin A+cos A)/(sin A-cos A)+(sin A-cos A)/(sin A+cos A)=(2)/(2sin^(2)A-1)

55. sin⁡A+cos⁡Asin⁡A−cos⁡A+sin⁡A−cos⁡Asin⁡A+cos⁡A=22sin⁡2A−1 \frac{\sin A+\cos A}{\sin A-\cos A}+\frac{\sin A-\cos A}{\sin A+\cos A}=\frac{2}{2 \sin ^{2} A-1}

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Full solution

Q. 55. sin⁡A+cos⁡Asin⁡A−cos⁡A+sin⁡A−cos⁡Asin⁡A+cos⁡A=22sin⁡2A−1 \frac{\sin A+\cos A}{\sin A-\cos A}+\frac{\sin A-\cos A}{\sin A+\cos A}=\frac{2}{2 \sin ^{2} A-1}
  1. Find Common Denominator: Simplify the left-hand side of the equation by finding a common denominator for the two fractions.
  2. Expand Numerators: Expand the numerators using the identity a2+b2=2a2+2b2−(a−b)2a^2 + b^2 = 2a^2 + 2b^2 - (a - b)^2.
  3. Use Pythagorean Identity: Use the Pythagorean identity sin2A+cos2A=1sin^2 A + cos^2 A = 1.
  4. Simplify Denominator: Simplify the denominator using the difference of squares formula.
  5. Substitute Numerator and Denominator: Substitute the simplified numerator and denominator back into the equation.
  6. Rewrite Denominator: Recognize that the denominator can be rewritten using the identity cos2A=cos2A−sin2Acos 2A = cos^2 A - sin^2 A.

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