We Trigonometry identities to verification

\(\displaystyle{\sec{\theta}}{\cot{\theta}}-{\sin{\theta}}={\frac{{{{\cos}^{{2}}\theta}}}{{{\sin{\theta}}}}}\)

\(\displaystyle{\sec{\theta}}{\cot{\theta}}-{\sin{\theta}}={\frac{{{1}}}{{{\cos{\theta}}}}}\times{\frac{{{\cos{\theta}}}}{{{\sin{\theta}}}}}-{\sin{\theta}}\)

\(\displaystyle\because{\sec{\theta}}={\frac{{{1}}}{{{\cos{\theta}}}}}\)

\(\displaystyle\because{\cot{\theta}}={\frac{{{\cos{\theta}}}}{{{\sin{\theta}}}}}\)

\(\displaystyle={\frac{{{1}}}{{{\sin{\theta}}}}}-{\frac{{{\sin{\theta}}}}{{{1}}}}\)

\(\displaystyle={\frac{{{1}-{{\sin}^{{2}}\theta}}}{{{\sin{\theta}}}}}\)

\(\displaystyle{\sec{\theta}}{\cot{\theta}}-{\sin{\theta}}={\frac{{{{\cos}^{{2}}\theta}}}{{{\sin{\theta}}}}}\)

\(\displaystyle{\sec{\theta}}{\cot{\theta}}-{\sin{\theta}}={\frac{{{{\cos}^{{2}}\theta}}}{{{\sin{\theta}}}}}\)

\(\displaystyle{\sec{\theta}}{\cot{\theta}}-{\sin{\theta}}={\frac{{{1}}}{{{\cos{\theta}}}}}\times{\frac{{{\cos{\theta}}}}{{{\sin{\theta}}}}}-{\sin{\theta}}\)

\(\displaystyle\because{\sec{\theta}}={\frac{{{1}}}{{{\cos{\theta}}}}}\)

\(\displaystyle\because{\cot{\theta}}={\frac{{{\cos{\theta}}}}{{{\sin{\theta}}}}}\)

\(\displaystyle={\frac{{{1}}}{{{\sin{\theta}}}}}-{\frac{{{\sin{\theta}}}}{{{1}}}}\)

\(\displaystyle={\frac{{{1}-{{\sin}^{{2}}\theta}}}{{{\sin{\theta}}}}}\)

\(\displaystyle{\sec{\theta}}{\cot{\theta}}-{\sin{\theta}}={\frac{{{{\cos}^{{2}}\theta}}}{{{\sin{\theta}}}}}\)