(tan alpha+cot alpha)/(tan alpha-cot alpha)=(1)/(2sin^(2)alpha-1)

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Trigonometric Identities Substitution: Recall the trigonometric identities for tangent and cotangent in terms of sine and cosine: tan(alpha) = sin(alpha)/cos(alpha) and cot(alpha) = cos(alpha)/sin(alpha). Substitute these identities into the left side of the equation.Combine Terms: The left side of the equation becomes: (sin(alpha)/cos(alpha) + cos(alpha)/sin(alpha)) / (sin(alpha)/cos(alpha) - cos(alpha)/sin(alpha)). Combine the terms over a common denominator.Simplify Using Pythagorean Identity: After combining terms, the left side simplifies to: (sin^2(alpha) + cos^2(alpha)) / (sin(alpha)cos(alpha)) / ((sin^2(alpha) - cos^2(alpha)) / (sin(alpha)cos(alpha))). Simplify the numerator and denominator using the Pythagorean identity sin^2(alpha) + cos^2(alpha) = 1.Complex Fraction Simplification: The left side now simplifies to: 1 / (sin(alpha)cos(alpha)) / ((sin^2(alpha) - cos^2(alpha)) / (sin(alpha)cos(alpha))). Simplify the complex fraction by multiplying the numerator and denominator by sin(alpha)cos(alpha).Double Angle Identity Substitution: The left side simplifies further to: 1 / (sin^2(alpha) - cos^2(alpha)). Now, recall the double angle identity for sine: sin^2(alpha) = (1 - cos(2alpha))/2. Substitute this identity into the denominator.Use Pythagorean Identity for Cosine: The left side becomes: 1 / ((1 - cos(2alpha))/2 - cos^2(alpha)). Now, use the Pythagorean identity cos^2(alpha) = 1 - sin^2(alpha) to express cos^2(alpha) in terms of sin^2(alpha).Denominator Simplification: The left side now becomes: 1 / ((1 - cos(2alpha))/2 - (1 - sin^2(alpha))). Simplify the denominator.Final Simplification: After simplifying, the left side is: 1 / ((1 - cos(2alpha) - 2 + 2sin^2(alpha))/2). This simplifies to: 1 / (2sin^2(alpha) - 1 - cos(2alpha))/2.Comparison with Right Side: Now, we need to show that the right side of the original equation, 1 / (2sin^2(alpha) - 1), is equal to the left side. However, we have an extra term, -cos(2alpha)/2, in the denominator of the left side that does not appear in the right side. This indicates that there might be a mistake in our simplification or the identity does not hold true.

$\frac{\tan \alpha+\cot \alpha}{\tan \alpha-\cot \alpha}=\frac{1}{2 \sin ^{2} \alpha-1}$

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$\frac{\tan \alpha+\cot \alpha}{\tan \alpha-\cot \alpha}=\frac{1}{2 \sin ^{2} \alpha-1}$