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

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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Sin, cos, and tan of special anglesright chevron icon

Full solution

Q. 1+sec⁡Asec⁡A=sin⁡2A1−cos⁡A \frac{1+\sec A}{\sec A}=\frac{\sin ^{2} A}{1-\cos A}
  1. Recognize sec⁡A\sec A Calculation: Recognize that sec⁡A\sec A is 1cos⁡A\frac{1}{\cos A}.\newlineCalculation: sec⁡A=1cos⁡A\sec A = \frac{1}{\cos A}.
  2. Rewrite using sec⁡A\sec A: Rewrite the left side of the equation using sec⁡A=1cos⁡A\sec A = \frac{1}{\cos A}.\newlineCalculation: 1+sec⁡Asec⁡A=1+1cos⁡A1cos⁡A\frac{1+\sec A}{\sec A} = \frac{1 + \frac{1}{\cos A}}{\frac{1}{\cos A}}.
  3. Combine over common denominator: Combine terms over a common denominator on the left side.\newlineCalculation: (1+1cos⁡A)/(1cos⁡A)=(cos⁡A+1)/(cos⁡A∗(1cos⁡A))(1 + \frac{1}{\cos A})/(\frac{1}{\cos A}) = (\cos A + 1)/(\cos A * (\frac{1}{\cos A})).
  4. Simplify by multiplying: Simplify the left side by multiplying the numerator and the denominator by cos⁡A\cos A.\newlineCalculation: cos⁡A+1cos⁡A⋅(1/cos⁡A)=cos⁡A+11\frac{\cos A + 1}{\cos A \cdot (1/\cos A)} = \frac{\cos A + 1}{1}.
  5. Further simplify: Further simplify the left side.\newlineCalculation: (cos⁡A+1)/1=cos⁡A+1(\cos A + 1)/1 = \cos A + 1.
  6. Recognize incorrect simplification: Recognize that cos⁡A+1\cos A + 1 is not the correct simplification; we need to cancel out cos⁡A\cos A in the numerator and denominator.\newlineCalculation: (cos⁡A+1)/1(\cos A + 1)/1 should be simplified to cos⁡A+1\cos A + 1, but this is incorrect.

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