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Alex Ar [27]
3 years ago
11

1+secA/sec A = sin^2 A / 1-cos A​

Mathematics
1 answer:
Fofino [41]3 years ago
3 0

Answer:  see proof below

<u>Step-by-step explanation:</u>

\dfrac{1+\sec A}{\sec A}=\dfrac{\sin^2 A}{1-\cos A}

Use the following Identities:

sec Ф = 1/cos Ф

cos² Ф + sin² Ф = 1

<u>Proof LHS → RHS</u>

\text{LHS:}\qquad \qquad \dfrac{1+\sec A}{\sec A}

\text{Identity:}\qquad \qquad \dfrac{1+\frac{1}{\cos A}}{\frac{1}{\cos A}}

\text{Simplify:}\qquad \qquad \dfrac{\frac{\cos A+1}{\cos A}}{\frac{1}{\cos A}}\\\\\\.\qquad \qquad \qquad =\dfrac{1+\cos A}{1}

\text{Multiply:}\qquad \qquad \dfrac{1+\cos A}{1}\cdot \bigg(\dfrac{1-\cos A}{1-\cos A}\bigg)\\\\\\.\qquad \qquad \qquad =\dfrac{1-\cos^2 A}{1-\cos A}

\text{Identity:}\qquad \qquad \dfrac{\sin^2 A}{1-\cos A}

\text{LHS = RHS:}\quad \dfrac{\sin^2 A}{1-\cos A}=\dfrac{\sin^2 A}{1-\cos A}\quad \checkmark

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