Solve dy=sin^2θ+cos^2θ/1+tan^2θdθ

$$d y = \frac { \sin ^ { 2 } \theta + \cos ^ { 2 } \theta } { 1 + \tan ^ { 2 } \theta } d \theta$$

Solve for y

$$\left\{\begin{matrix}y=\theta \left(\cos(\theta )\right)^{2}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\\y\in \mathrm{R}\text{, }&d=0\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Express \frac{\left(\sin(\theta )\right)^{2}+\left(\cos(\theta )\right)^{2}}{1+\left(\tan(\theta )\right)^{2}}d as a single fraction.

$$dy=\frac{\left(\left(\sin(\theta )\right)^{2}+\left(\cos(\theta )\right)^{2}\right)d}{1+\left(\tan(\theta )\right)^{2}}\theta$$

Express \frac{\left(\left(\sin(\theta )\right)^{2}+\left(\cos(\theta )\right)^{2}\right)d}{1+\left(\tan(\theta )\right)^{2}}\theta as a single fraction.

$$dy=\frac{\left(\left(\sin(\theta )\right)^{2}+\left(\cos(\theta )\right)^{2}\right)d\theta }{1+\left(\tan(\theta )\right)^{2}}$$

Use the distributive property to multiply \left(\sin(\theta )\right)^{2}+\left(\cos(\theta )\right)^{2} by d.

$$dy=\frac{\left(\left(\sin(\theta )\right)^{2}d+\left(\cos(\theta )\right)^{2}d\right)\theta }{1+\left(\tan(\theta )\right)^{2}}$$

Use the distributive property to multiply \left(\sin(\theta )\right)^{2}d+\left(\cos(\theta )\right)^{2}d by \theta .

$$dy=\frac{\left(\sin(\theta )\right)^{2}d\theta +\left(\cos(\theta )\right)^{2}d\theta }{1+\left(\tan(\theta )\right)^{2}}$$

The equation is in standard form.

$$dy=\frac{d\theta \left(\sin(\theta )\right)^{2}+d\theta \left(\cos(\theta )\right)^{2}}{\left(\tan(\theta )\right)^{2}+1}$$

Divide both sides by d.

$$\frac{dy}{d}=\frac{d\theta \left(\cos(\theta )\right)^{2}}{d}$$

Dividing by d undoes the multiplication by d.

$$y=\frac{d\theta \left(\cos(\theta )\right)^{2}}{d}$$

Divide d\theta \left(\cos(\theta )\right)^{2} by d.

$$y=\theta \left(\cos(\theta )\right)^{2}$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\pi n_{1}+\frac{\pi }{2}\\d\in \mathrm{R}\text{, }&y=\theta \left(\cos(\theta )\right)^{2}\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\theta =\frac{\pi \left(2n_{1}+1\right)}{2}\end{matrix}\right.$$