Solve 1=2(10rsin2θsinθcosphi)

$$1 = 2 ( 10 r \sin 2 \theta \sin \theta \cos \phi )$$

Solve for r

$$r=\frac{1}{40\cos(\theta )\cos(ϕ)\left(\sin(\theta )\right)^{2}}$$

$$\nexists n_{1}\in \mathrm{Z}\text{ : }ϕ=\pi n_{1}+\frac{\pi }{2}\text{ and }\nexists n_{2}\in \mathrm{Z}\text{ : }\theta =\frac{\pi n_{2}}{2}$$

Steps for Solving Linear Equation

Multiply 2 and 10 to get 20.

$$1=20r\sin(2\theta )\sin(\theta )\cos(ϕ)$$

Swap sides so that all variable terms are on the left hand side.

$$20r\sin(2\theta )\sin(\theta )\cos(ϕ)=1$$

The equation is in standard form.

$$20\sin(\theta )\sin(2\theta )\cos(ϕ)r=1$$

Divide both sides by 20\sin(2\theta )\sin(\theta )\cos(ϕ).

$$\frac{20\sin(\theta )\sin(2\theta )\cos(ϕ)r}{20\sin(\theta )\sin(2\theta )\cos(ϕ)}=\frac{1}{20\sin(\theta )\sin(2\theta )\cos(ϕ)}$$

Dividing by 20\sin(2\theta )\sin(\theta )\cos(ϕ) undoes the multiplication by 20\sin(2\theta )\sin(\theta )\cos(ϕ).

$$r=\frac{1}{20\sin(\theta )\sin(2\theta )\cos(ϕ)}$$

Divide 1 by 20\sin(2\theta )\sin(\theta )\cos(ϕ).

$$r=\frac{1}{40\cos(\theta )\cos(ϕ)\left(\sin(\theta )\right)^{2}}$$