How to simplify cot theta sec theta sin theta?

Kristin Montgomery

Kristin Montgomery

Answered question

2023-01-04

How to simplify cot⁡θsec⁡θsin⁡θ?

Answer & Explanation

Armeebildzku

Armeebildzku

Beginner2023-01-05Added 14 answers

Use the following two identities:
\(\displaystyle{\cot{\theta}}=\frac{{1}}{{\tan{\theta}}}=\frac{{1}}{{\frac{{\sin{\theta}}}{{\cos{\theta}}}}}=\frac{{\cos{\theta}}}{{\sin{\theta}}}\)
\(\displaystyle{\sec{\theta}}=\frac{{1}}{{\cos{\theta}}}\)
The expression is as follows:
\(\displaystyle{\color{white}=}{\cot{\theta}}{\sec{\theta}}{\sin{\theta}}\)
\(\displaystyle={\cot{\theta}}\cdot{\sec{\theta}}\cdot{\sin{\theta}}\)
\(\displaystyle=\frac{{\cos{\theta}}}{{\sin{\theta}}}\cdot\frac{{1}}{{\cos{\theta}}}\cdot{\sin{\theta}}\)
\(\displaystyle=\frac{{\color{red}\cancel{{\color{black}{\cos{\theta}}}}}}{{\sin{\theta}}}\cdot\frac{{1}}{{\color{red}\cancel{{\color{black}{\cos{\theta}}}}}}\cdot{\sin{\theta}}\)
\(\displaystyle=\frac{{1}}{{\sin{\theta}}}\cdot{1}\cdot{\sin{\theta}}\)
\(\displaystyle=\frac{{1}}{{\sin{\theta}}}\cdot{\sin{\theta}}\)
\(\displaystyle=\frac{{1}}{{\color{red}\cancel{{\color{black}{\sin{\theta}}}}}}\cdot{\color{red}\cancel{{\color{black}{\sin{\theta}}}}}\)
\(\displaystyle={1}\)

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