Verify the identify.\frac{\tan^2\theta}{\sec\theta+1}=\frac{1-\cos x}{\cos x}

coexpennan 2021-09-12 Answered
Verify the identify.
\(\displaystyle{\frac{{{{\tan}^{{2}}\theta}}}{{{\sec{\theta}}+{1}}}}={\frac{{{1}-{\cos{{x}}}}}{{{\cos{{x}}}}}}\)

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Expert Answer

Laith Petty
Answered 2021-09-13 Author has 5786 answers
Solution:
\(\displaystyle{L}{H}{S}={\frac{{{{\tan}^{{2}}\theta}}}{{{\sec{\theta}}+{1}}}}\)
\(\displaystyle={\frac{{{\left({\frac{{{\sin{\theta}}}}{{{\cos{\theta}}}}}\right)}^{{2}}}}{{{\frac{{{1}}}{{{\cos{\theta}}}}}+{1}}}}\)
\(\displaystyle={\frac{{{\frac{{{{\sin}^{{2}}\theta}}}{{{{\cos}^{{2}}\theta}}}}}}{{{\frac{{{1}+{\cos{\theta}}}}{{{\cos{\theta}}}}}}}}\)
\(\displaystyle={\frac{{{{\sin}^{{2}}\theta}}}{{{\cos{\theta}}{\left({1}+{\cos{\theta}}\right)}}}}\)
\(\displaystyle={\frac{{{\left({1}-{{\cos}^{{2}}\theta}\right)}}}{{{\cos{\theta}}{\left({1}+{\cos{\theta}}\right)}}}}\)
\(\displaystyle={\frac{{{\left({1}-{\cos{\theta}}\right)}{\left({1}+{\cos{\theta}}\right)}}}{{{\cos{\theta}}{\left({1}+{\cos{\theta}}\right)}}}}\)
\(\displaystyle={\frac{{{1}-{\cos{\theta}}}}{{{\cos{\theta}}}}}\)
\(\displaystyle={R}{H}{S}\)
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