Question

Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4).

Answer 1

$$\begin{gathered} f\left(x\right)=\sin x-\cos x \\ f\text{'}\left(x\right)=\cos x+\sin x \\ =\cos x\left(1+\frac{\sin x}{\cos x}\right) \\ =\cos x\left(1+\cot x\right) \\ \mathrm{Here}, \\ \frac{-\mathrm{\pi }}{4}<x<\frac{\mathrm{\pi }}{4} \\ \Rightarrow \cos x>0...\left(1\right) \\ \mathrm{Also}, \\ \frac{-\mathrm{\pi }}{4}<x<\frac{\mathrm{\pi }}{4}\Rightarrow -1<\cot x<1 \\ \Rightarrow 0<1+\cot x<2 \\ \Rightarrow 1+\cot x>0...\left(2\right) \\ \cos x\left(1+\cot x\right)>0,\forall x\in \left(\frac{-\mathrm{\pi }}{4},\frac{\mathrm{\pi }}{4}\right)\left[\text{From eqs. (1) and (2)}\right] \\ \Rightarrow f\text{'}\left(x\right)>0,\forall x\in \left(\frac{-\mathrm{\pi }}{4},\frac{\mathrm{\pi }}{4}\right) \\ \text{So,}f\left(x\right)\text{is increasing on}\left(\frac{-\mathrm{\pi }}{4},\frac{\mathrm{\pi }}{4}\right)\text{.} \end{gathered}$$