( 1/(1- sin x) + 1/(1+ sin x) )+( ( cos x)/(1+ sin x) + tan x)

asked by guest
on Oct 19, 2024 at 12:29 pm



You asked:

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

MathBot Answer:

Evaluated



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

Expanded

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

Factored

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

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