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Prove the identities with all angles acute i. (1+sec A)/(sec A)=(sin^(2)A)/(1-cos A)

Prove the identities with all angles acute\newlinei. 1+sec⁡Asec⁡A=sin⁡2A1−cos⁡A \frac{1+\sec A}{\sec A}=\frac{\sin ^{2} A}{1-\cos A}

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Q. Prove the identities with all angles acute\newlinei. 1+sec⁡Asec⁡A=sin⁡2A1−cos⁡A \frac{1+\sec A}{\sec A}=\frac{\sin ^{2} A}{1-\cos A}
  1. Rewrite using secant definition: Rewrite the left side of the equation using the definition of secant: sec⁡A=1cos⁡A\sec A = \frac{1}{\cos A}. So, (1+sec⁡A)/(sec⁡A)(1+\sec A)/(\sec A) becomes (1+1cos⁡A)/(1cos⁡A)(1 + \frac{1}{\cos A})/(\frac{1}{\cos A}).
  2. Combine terms in numerator: Combine the terms in the numerator: 1+1cos⁡A=cos⁡A+1cos⁡A.1 + \frac{1}{\cos A} = \frac{\cos A + 1}{\cos A}.
  3. Divide numerator by denominator: Now, divide the combined numerator by the denominator: (cos⁡A+1cos⁡A)/(1cos⁡A)\left(\frac{\cos A + 1}{\cos A}\right) / \left(\frac{1}{\cos A}\right).
  4. Simplify division by reciprocal: Simplify the division by multiplying by the reciprocal of the denominator: (cos⁡A+1)/cos⁡A×cos⁡A/1(\cos A + 1)/\cos A \times \cos A/1.
  5. Cancel out cos⁡A\cos A terms: The cos⁡A\cos A terms cancel out, leaving us with cos⁡A+1\cos A + 1.
  6. Use Pythagorean identity: Now, let's look at the right side of the equation: sin⁡2A1−cos⁡A\frac{\sin^{2}A}{1-\cos A}.
  7. Substitute identity into right side: Use the Pythagorean identity sin⁡2A=1−cos⁡2A\sin^2 A = 1 - \cos^2 A.
  8. Factor numerator as difference of squares: Substitute the identity into the right side of the equation: (1−cos⁡2A)/(1−cos⁡A)(1 - \cos^2 A)/(1 - \cos A).
  9. Cancel out (1−cos⁡A)(1 - \cos A) terms: Factor the numerator as a difference of squares: (1−cos⁡A)(1+cos⁡A)1−cos⁡A\frac{(1 - \cos A)(1 + \cos A)}{1 - \cos A}.
  10. Final expression on both sides: The (1−cos⁡A)(1 - \cos A) terms cancel out, leaving us with 1+cos⁡A1 + \cos A.
  11. Final expression on both sides: The (1−cos⁡A)(1 - \cos A) terms cancel out, leaving us with 1+cos⁡A1 + \cos A.We now have the same expression on both sides of the equation: cos⁡A+1\cos A + 1.

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