What is the simplified value of the expression \(\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}\) ?
cos A + sin A
The given expression is:
\(\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}\)
We need to simplify this expression. We start by looking at the trigonometric identities:
- Tangent: \(\tan A = \frac{\sin A}{\cos A}\)
- Cotangent: \(\cot A = \frac{\cos A}{\sin A}\)
Rewriting the given expression using these identities:
\(\frac{\cos A}{1-\frac{\sin A}{\cos A}} + \frac{\sin A}{1-\frac{\cos A}{\sin A}}\)
Simplify each fraction:
\(\frac{\cos A}{1-\frac{\sin A}{\cos A}} = \frac{\cos^2 A}{\cos A - \sin A}\)
\(\frac{\sin A}{1-\frac{\cos A}{\sin A}} = \frac{\sin^2 A}{\sin A - \cos A}\)
Now, add the two resulting fractions:
\(\frac{\cos^2 A}{\cos A - \sin A} + \frac{\sin^2 A}{\sin A - \cos A}\)
Combine the terms over a common denominator, \((\cos A - \sin A)(\sin A - \cos A)\):
\(\frac{\cos^2 A (\sin A - \cos A) + \sin^2 A (\cos A - \sin A)}{(\cos A - \sin A)(\sin A - \cos A)}\)
Notice the numerator simplifies using the identity \(a^2 - b^2 = (a+b)(a-b)\):
\(\cos^2 A \cdot \sin A - \cos^3 A + \sin^2 A \cdot \cos A - \sin^3 A = (\cos A + \sin A)(\cos A - \sin A)\)
The denominator becomes simple as it is a difference product:
\((\cos A - \sin A)(-(\cos A - \sin A)) = -(\cos A - \sin A)^2\)
Thus, the expression becomes:
\(\frac{(\cos A + \sin A)(\cos A - \sin A)}{-(\cos A - \sin A)^2} = \frac{-(\cos^2 A - \sin^2 A)}{-(\cos A - \sin A)^2} = \cos A + \sin A\)
Thus, the simplified value of the expression is:
cos A + sin A
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