Question

Verify the identify.\frac{1+\csc3\beta}{\sec3\beta}-\cot3\beta=\cos3\beta

Trigonometric equation and identitie
ANSWERED
asked 2021-09-11
Verify the identify.
\(\displaystyle{\frac{{{1}+{\csc{{3}}}\beta}}{{{\sec{{3}}}\beta}}}-{\cot{{3}}}\beta={\cos{{3}}}\beta\)

Expert Answers (1)

2021-09-12
Solution:
\(\displaystyle{L}{H}{S}={\frac{{{1}+{\csc{{3}}}\beta}}{{{\sec{{3}}}\beta}}}-{\cot{{3}}}\beta\)
\(\displaystyle={\frac{{{1}+{\frac{{{1}}}{{{\sin{{3}}}\beta}}}}}{{{\frac{{{1}}}{{{\cos{{3}}}\beta}}}}}}-{\frac{{{\cos{{3}}}\beta}}{{{\sin{{3}}}\beta}}}\)
\(\displaystyle={\frac{{{\frac{{{\sin{{3}}}\beta+{1}}}{{{\sin{{3}}}\beta}}}}}{{{\frac{{{1}}}{{{\cos{{3}}}\beta}}}}}}-{\frac{{{\cos{{3}}}\beta}}{{{\sin{{3}}}\beta}}}\)
\(\displaystyle={\frac{{{\cos{{3}}}\beta}}{{{\sin{{3}}}\beta}}}{\left({\sin{{3}}}\beta+{1}\right)}-{\frac{{{\cos{{3}}}\beta}}{{{\sin{{3}}}\beta}}}\)
\(\displaystyle={\frac{{{\cos{{3}}}\beta{\left[{\sin{{3}}}\beta+{1}-{1}\right]}}}{{{\sin{{3}}}\beta}}}\)
\(\displaystyle={\cos{{3}}}\beta{\frac{{{\left[{\sin{{3}}}\beta\right]}}}{{{\sin{{3}}}\beta}}}\)
\(\displaystyle={\cos{{3}}}\beta\)
\(\displaystyle={R}{H}{S}\)
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