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BabaBlast [244]
3 years ago
5

How would you prove this trigonomic identity? Please show steps.

Mathematics
1 answer:
Agata [3.3K]3 years ago
4 0

Start by combining the fractions:

\dfrac{\cos\alpha}{1+\sin\alpha}\cdot\dfrac{1-\sin\alpha}{1-\sin\alpha}+\dfrac{\cos\alpha}{1-\sin\alpha}\cdot\dfrac{1+\sin\alpha}{1+\sin\alpha}

\dfrac{\cos\alpha(1-\sin\alpha)+\cos\alpha(1+\sin\alpha)}{(1+\sin\alpha)(1-\sin\alpha)}

\dfrac{2\cos\alpha}{1-\sin^2\alpha}

Recall the Pythagorean identity:

\dfrac{2\cos\alpha}{\cos^2\alpha}

Then cancel a factor of \cos\alpha and use the definition of secant:

\dfrac2{\cos\alpha}=\boxed{2\sec\alpha}

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