\(\left(\frac{\sin x \cdot \cos \varepsilon}{\cos \left(\varepsilon + x\right)} + \frac{\cos x \cdot \sin \varepsilon}{\cos \left(\varepsilon + x\right)}\right) - \tan x\)
- Started with
\[\tan \left(x + \varepsilon\right) - \tan x\]
37.4
- Using strategy
rm 37.4
- Applied tan-cotan to get
\[\color{red}{\tan \left(x + \varepsilon\right)} - \tan x \leadsto \color{blue}{\frac{1}{\cot \left(x + \varepsilon\right)}} - \tan x\]
37.4
- Using strategy
rm 37.4
- Applied cotan-quot to get
\[\frac{1}{\color{red}{\cot \left(x + \varepsilon\right)}} - \tan x \leadsto \frac{1}{\color{blue}{\frac{\cos \left(x + \varepsilon\right)}{\sin \left(x + \varepsilon\right)}}} - \tan x\]
37.5
- Applied associate-/r/ to get
\[\color{red}{\frac{1}{\frac{\cos \left(x + \varepsilon\right)}{\sin \left(x + \varepsilon\right)}}} - \tan x \leadsto \color{blue}{\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \sin \left(x + \varepsilon\right)} - \tan x\]
37.4
- Using strategy
rm 37.4
- Applied sin-sum to get
\[\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \color{red}{\sin \left(x + \varepsilon\right)} - \tan x \leadsto \frac{1}{\cos \left(x + \varepsilon\right)} \cdot \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \tan x\]
36.5
- Applied distribute-lft-in to get
\[\color{red}{\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \tan x \leadsto \color{blue}{\left(\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\sin x \cdot \cos \varepsilon\right) + \frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\cos x \cdot \sin \varepsilon\right)\right)} - \tan x\]
36.5
- Applied simplify to get
\[\left(\color{red}{\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\sin x \cdot \cos \varepsilon\right)} + \frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\cos x \cdot \sin \varepsilon\right)\right) - \tan x \leadsto \left(\color{blue}{\frac{\sin x \cdot \cos \varepsilon}{\cos \left(\varepsilon + x\right)}} + \frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\cos x \cdot \sin \varepsilon\right)\right) - \tan x\]
36.4
- Applied simplify to get
\[\left(\frac{\sin x \cdot \cos \varepsilon}{\cos \left(\varepsilon + x\right)} + \color{red}{\frac{1}{\cos \left(x + \varepsilon\right)} \cdot \left(\cos x \cdot \sin \varepsilon\right)}\right) - \tan x \leadsto \left(\frac{\sin x \cdot \cos \varepsilon}{\cos \left(\varepsilon + x\right)} + \color{blue}{\frac{\cos x \cdot \sin \varepsilon}{\cos \left(\varepsilon + x\right)}}\right) - \tan x\]
36.4
