Question

Simplify the expression secx/sinx - sinx/cosx

Trigonometric equation and identitie
ANSWERED
asked 2020-11-01
Simplify the expression \(\displaystyle\frac{{\sec{{x}}}}{{\sin{{x}}}}-\frac{{\sin{{x}}}}{{\cos{{x}}}}\)

Answers (1)

2020-11-02
\(\displaystyle{\sec{{x}}}=\frac{{1}}{{\cos{{x}}}}\)
\(\displaystyle\frac{{\frac{{1}}{{\cos{{x}}}}}}{{\sin{{x}}}}-\frac{{\cos{{x}}}}{{\sin{{x}}}}\)
\(\displaystyle\frac{{\frac{{1}}{{\cos{{x}}}}-{\cos{{x}}}}}{{\sin{{x}}}}\)
\(\displaystyle\frac{{\frac{{1}}{{\cos{{x}}}}-\frac{{{\cos}^{{2}}{x}}}{{\cos{{x}}}}}}{{\sin{{x}}}}\)
\(\displaystyle\frac{{{1}-\frac{{{\cos}^{{2}}{x}}}{{\cos{{x}}}}}}{{\sin{{x}}}}\)
\(\displaystyle{{\cos}^{{2}}{x}}+{{\sin}^{{2}}{x}}={1}\rightarrow{{\sin}^{{2}}{x}}={1}-{{\cos}^{{2}}{x}}\)
\(\displaystyle\frac{{\frac{{{\sin}^{{2}}{x}}}{{\cos{{x}}}}}}{{\sin{{x}}}}\rightarrow{\left(\frac{{{\sin}^{{2}}{x}}}{{\cos{{x}}}}\right)}•{\left(\frac{{1}}{{\sin{{x}}}}\right)}\)
\(\displaystyle=\frac{{{{\sin}^{{2}}{x}}}}{{\sin{{x}}}}{\cos{{x}}}\)
\(\displaystyle\frac{{\sin{{x}}}}{{\cos{{x}}}}={\tan{{x}}}\)
\(\displaystyle={\tan{{x}}}\)
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