Solve 2tan2(1+operatorname{ctan}2)^2

$$2 \tan 2 ( 1 + \operatorname { ctan } 2 ) ^ { 2 }$$

Evaluate

$$\frac{2\tan(2)\left(\sin(2)c+\cos(2)\right)^{2}}{\left(\cos(2)\right)^{2}}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+c\tan(2)\right)^{2}.

$$2\tan(2)\left(1+2c\tan(2)+c^{2}\left(\tan(2)\right)^{2}\right)$$

Use the distributive property to multiply 2\tan(2) by 1+2c\tan(2)+c^{2}\left(\tan(2)\right)^{2}.

$$2\tan(2)+4\left(\tan(2)\right)^{2}c+2c^{2}\left(\tan(2)\right)^{3}$$

Differentiate w.r.t. c

$$\frac{4\left(\tan(2)\right)^{2}\left(\sin(2)c+\cos(2)\right)}{\cos(2)}$$