$\tan 2 u=\frac{2 \cot u}{\csc ^{2} u-2}$

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Algebra 2

Sin, cos, and tan of special angles

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\tan 2 u=\frac{2 \cot u}{\csc ^{2} u-2}

Recall Identities: Recall the trigonometric identities: $\cot u = \frac{1}{\tan u}$ and $\csc u = \frac{1}{\sin u}.$
Rewrite Equation: Rewrite the equation using the identities: $\tan 2u = \frac{2 \times \left(\frac{1}{\tan u}\right)}{\left(\frac{1}{\sin u}\right)^{2} - 2}$ .
Simplify Equation: Simplify the equation: $\tan 2u = \frac{2/\tan u}{(1/\sin^{2}u) - 2}$ .
Convert to sin/cos: Convert $\tan u$ to $\sin u$ and $\cos u$ : $\tan 2u = \frac{2 \times \left(\frac{\cos u}{\sin u}\right)}{\left(\frac{1}{\sin^{2}u}\right) - 2}$ .
Clear Fractions: Multiply both numerator and denominator by $\sin^{2}u$ to clear the fraction: $\tan 2u = \frac{2 \times (\cos u/\sin u) \times \sin^{2}u}{1 - 2\sin^{2}u}$ .
Use Double Angle Identity: Simplify the equation: $\tan 2u = \frac{2\cos u \cdot \sin u}{1 - 2\sin^{2}u}.$
Substitute $\sin(2u)$ : Use the double angle identity for sine: $\sin(2u) = 2\sin u \cdot \cos u$ .
Use Pythagorean Identity: Substitute $\sin(2u)$ into the equation: $\tan 2u = \frac{\sin(2u)}{(1 - 2\sin^{2}u)}$ .
Find $\cos^2 u$ : Use the Pythagorean identity: $\sin^2 u + \cos^2 u = 1$ .
Substitute $\cos^2 u$ : Rearrange the identity to find $\cos^2 u$ : $\cos^2 u = 1 - \sin^2 u$ .
Final Simplification: Substitute $\cos^2 u$ into the equation: $\tan 2u = \frac{\sin(2u)}{1 - 2(1 - \cos^2 u)}$ .
Use Double Angle Identity: Simplify the equation: $\tan 2u = \frac{\sin(2u)}{2\cos^{2}u - 1}.$
Recognize Mistake: Use the double angle identity for cosine: $\cos(2u) = 2\cos^2(u) - 1$ .
Recognize Mistake: Use the double angle identity for cosine: $\cos(2u) = 2\cos^2(u) - 1$ . Substitute $\cos(2u)$ into the equation: $\tan 2u = \frac{\sin(2u)}{\cos(2u)}$ .
Recognize Mistake: Use the double angle identity for cosine: $\cos(2u) = 2\cos^2(u) - 1$ .Substitute $\cos(2u)$ into the equation: $\tan 2u = \frac{\sin(2u)}{\cos(2u)}$ .Recognize that $\tan 2u = \frac{\sin(2u)}{\cos(2u)}$ is the definition of tangent: $\tan 2u = \tan(2u)$ .
Recognize Mistake: Use the double angle identity for cosine: $\cos(2u) = 2\cos^2(u) - 1$ . Substitute $\cos(2u)$ into the equation: $\tan 2u = \frac{\sin(2u)}{\cos(2u)}$ . Recognize that $\tan 2u = \frac{\sin(2u)}{\cos(2u)}$ is the definition of tangent: $\tan 2u = \tan(2u)$ . Realize that we've made a mistake in the simplification process; the correct simplification should have led to $\tan 2u = \tan(2u)$ directly, without the intermediate steps.