Solve cot(90-alpha)

$$\cot ( 90 - \alpha )$$

Differentiate w.r.t. α

$$\frac{\left(\sin(90)\sin(\alpha )\right)^{2}+\left(\sin(90)\cos(\alpha )\right)^{2}+\left(\cos(90)\sin(\alpha )\right)^{2}+\left(\cos(90)\cos(\alpha )\right)^{2}}{\left(\sin(90-\alpha )\right)^{2}}$$

Steps Using Chain Rule

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$\left(-\left(\csc(-\alpha ^{1}+90)\right)^{2}\right)\frac{\mathrm{d}}{\mathrm{d}\alpha }(-\alpha ^{1}+90)$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$\left(-\left(\csc(-\alpha ^{1}+90)\right)^{2}\right)\left(-1\right)\alpha ^{1-1}$$

Simplify.

$$\left(-\left(-1\right)\right)\left(\csc(-\alpha ^{1}+90)\right)^{2}$$

For any term t, t^{1}=t.

$$\left(-\left(-1\right)\right)\left(\csc(-\alpha +90)\right)^{2}$$

Evaluate

$$\cot(90-\alpha )$$