$\frac{sina+cosa+1}{sina+cosa-1}$ - $\frac{1+cosa-cosa}{1-sina+cosa}$

asked by guest
on Nov 15, 2024 at 7:28 pm



You asked:

Evaluate the expression: \(\frac{\sin\left( a \right) + \cos\left( a \right) + 1}{\sin\left( a \right) + \cos\left( a \right) - 1} - \frac{1 + \cos\left( a \right) - \cos\left( a \right)}{1 - \sin\left( a \right) + \cos\left( a \right)}\)

MathBot Answer:

Evaluated



\(\displaystyle \frac{\sin\left( a \right) + \cos\left( a \right) + 1}{\sin\left( a \right) + \cos\left( a \right) - 1} - \frac{1 + \cos\left( a \right) - \cos\left( a \right)}{1 - \sin\left( a \right) + \cos\left( a \right)} = \frac{\sin{\left(a \right)} + \cos{\left(a \right)} + 1}{\sin{\left(a \right)} + \cos{\left(a \right)} - 1} - \frac{1}{- \sin{\left(a \right)} + \cos{\left(a \right)} + 1} \)

Expanded

\[\frac{\sin\left( a \right) + \cos\left( a \right) + 1}{\sin\left( a \right) + \cos\left( a \right) - 1} - \frac{1 + \cos\left( a \right) - \cos\left( a \right)}{1 - \sin\left( a \right) + \cos\left( a \right)} = \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} + \cos{\left(a \right)} - 1} + \frac{\cos{\left(a \right)}}{\sin{\left(a \right)} + \cos{\left(a \right)} - 1} + \frac{1}{\sin{\left(a \right)} + \cos{\left(a \right)} - 1} - \frac{1}{- \sin{\left(a \right)} + \cos{\left(a \right)} + 1}\]

Factored

\[\frac{\sin\left( a \right) + \cos\left( a \right) + 1}{\sin\left( a \right) + \cos\left( a \right) - 1} - \frac{1 + \cos\left( a \right) - \cos\left( a \right)}{1 - \sin\left( a \right) + \cos\left( a \right)} = \frac{- \sin^{2}{\left(a \right)} - \sin{\left(a \right)} + \cos^{2}{\left(a \right)} + \cos{\left(a \right)} + 2}{\left(- \sin{\left(a \right)} + \cos{\left(a \right)} + 1\right) \left(\sin{\left(a \right)} + \cos{\left(a \right)} - 1\right)}\]

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