$2$ . Verify the identity $\newline$ $\frac{\tan \alpha-1}{\tan \alpha+1}=\frac{1-\cot \alpha}{1+\cot \alpha}$

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Algebra 2

Simplify expressions using trigonometric identities

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2

. Verify the identity

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\frac{\tan \alpha-1}{\tan \alpha+1}=\frac{1-\cot \alpha}{1+\cot \alpha}

Express $\tan(\alpha)$ : Express $\tan(\alpha)$ in terms of $\sin(\alpha)$ and $\cos(\alpha)$ as $\tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)}$ .
Substitute $\tan(\alpha)$ : Substitute $\tan(\alpha)$ with $\frac{\sin(\alpha)}{\cos(\alpha)}$ in the given expression to get $\frac{\sin(\alpha)/\cos(\alpha) - 1}{\sin(\alpha)/\cos(\alpha) + 1}$ .
Find common denominator: Find a common denominator for the terms in the numerator and the denominator to combine them. This gives us $\left(\frac{\sin(\alpha) - \cos(\alpha)}{\cos(\alpha)}\right)/\left(\frac{\sin(\alpha) + \cos(\alpha)}{\cos(\alpha)}\right)$ .
Simplify complex fraction: Simplify the complex fraction by multiplying the numerator and the denominator by $\cos(\alpha)$ . This results in $\frac{\sin(\alpha) - \cos(\alpha)}{\sin(\alpha) + \cos(\alpha)}$ .
Express $\cot(\alpha)$ : Express $\cot(\alpha)$ in terms of $\cos(\alpha)$ and $\sin(\alpha)$ as $\cot(\alpha) = \frac{\cos(\alpha)}{\sin(\alpha)}$ .
Substitute $\cot(\alpha)$ : Substitute $\cot(\alpha)$ with $\frac{\cos(\alpha)}{\sin(\alpha)}$ in the right side of the identity to get $\frac{1 - \frac{\cos(\alpha)}{\sin(\alpha)}}{1 + \frac{\cos(\alpha)}{\sin(\alpha)}}$ .
Find common denominator: Find a common denominator for the terms in the numerator and the denominator to combine them. This gives us $\left(\frac{\sin(\alpha) - \cos(\alpha)}{\sin(\alpha)}\right)/\left(\frac{\sin(\alpha) + \cos(\alpha)}{\sin(\alpha)}\right)$ .
Simplify complex fraction: Simplify the complex fraction by multiplying the numerator and the denominator by $\sin(\alpha)$ . This results in $\frac{\sin(\alpha) - \cos(\alpha)}{\sin(\alpha) + \cos(\alpha)}$ .
Compare simplified expressions: Compare the simplified expressions from both sides of the identity. We have $(\sin(\alpha) - \cos(\alpha))/(\sin(\alpha) + \cos(\alpha))$ on both sides, which confirms the identity is true.