# What is the equation of the tangent line of #r=cot^2theta - sintheta# at #theta=pi/4#?

Then, we can just a point-slope formula to figure out the equation of that tangent line by plugging in values into the original function:

Thus:

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To find the equation of the tangent line to the curve ( r = \cot^2(\theta) - \sin(\theta) ) at the point ( \theta = \frac{\pi}{4} ), we first find the slope of the tangent line by taking the derivative of ( r ) with respect to ( \theta ) and then evaluate it at ( \theta = \frac{\pi}{4} ). Then we use the point-slope form of the equation of a line to find the equation of the tangent line.

Taking the derivative of ( r ) with respect to ( \theta ) yields:

[ \frac{dr}{d\theta} = \frac{d}{d\theta}(\cot^2(\theta) - \sin(\theta)) ]

[ = \frac{d}{d\theta}(\cot^2(\theta)) - \frac{d}{d\theta}(\sin(\theta)) ]

[ = -2\cot(\theta)\csc^2(\theta) - \cos(\theta) ]

Evaluating this derivative at ( \theta = \frac{\pi}{4} ) gives:

[ \left. \frac{dr}{d\theta} \right|_{\theta=\frac{\pi}{4}} = -2\cot\left(\frac{\pi}{4}\right)\csc^2\left(\frac{\pi}{4}\right) - \cos\left(\frac{\pi}{4}\right) ]

[ = -2(1)\left(\frac{\sqrt{2}}{2}\right)^2 - \frac{\sqrt{2}}{2} ]

[ = -1 - \frac{\sqrt{2}}{2} ]

Now, we have the slope of the tangent line. To find the equation of the tangent line, we use the point-slope form of a line with the point ( (\theta, r) = \left(\frac{\pi}{4}, \cot^2\left(\frac{\pi}{4}\right) - \sin\left(\frac{\pi}{4}\right)\right) ) and the slope we just found.

Substituting the values into the point-slope form, we get:

[ r - \left(\cot^2\left(\frac{\pi}{4}\right) - \sin\left(\frac{\pi}{4}\right)\right) = \left(-1 - \frac{\sqrt{2}}{2}\right)(\theta - \frac{\pi}{4}) ]

[ r - \left(1 - \frac{\sqrt{2}}{2}\right) = \left(-1 - \frac{\sqrt{2}}{2}\right)\left(\theta - \frac{\pi}{4}\right) ]

[ r = \left(-1 - \frac{\sqrt{2}}{2}\right)\left(\theta - \frac{\pi}{4}\right) + 1 - \frac{\sqrt{2}}{2} ]

This is the equation of the tangent line to the curve ( r = \cot^2(\theta) - \sin(\theta) ) at the point ( \theta = \frac{\pi}{4} ).

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