Question-75267

Question Number 75267 by aliesam last updated on 09/Dec/19

Commented by mind is power last updated on 09/Dec/19

let f(x)=(((√(1+sin(x)))+(√(1−sin(x))))/( (√(1+sin(x)))−(√(1−sin(x))))) sin(x)=2sin((x/2))cos((x/2)) 1+_− sin(x)=(cos((x/2))+_− sin((x/2)))^2 since x∈(0,(π/4))⇒(√(1−sin(x)))=∣cos((x/2))−sin((x/2))∣=cos((x/2))−sin((x/2)) ⇒(√(1+sin(x)))−(√(1−sin(x)))=2sin((x/2)) (√(1+sin(x)))+(√(1−sin(x)))=(cos((x/2))+sin((x/2))+cos((x/2))−sin((x/2)))=2cos((x/2)) f(x)=((2cos((x/2)))/(2sin((x/2))))=cot((x/2)) cot^− (f(x))=cot^− (cot((x/2)))=(x/2)

$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\sqrt{\mathrm{1}+\mathrm{sin}\left(\mathrm{x}\right)}+\sqrt{\mathrm{1}−\mathrm{sin}\left(\mathrm{x}\right)}}{\:\sqrt{\mathrm{1}+\mathrm{sin}\left(\mathrm{x}\right)}−\sqrt{\mathrm{1}−\mathrm{sin}\left(\mathrm{x}\right)}} \\ $$$$\mathrm{sin}\left(\mathrm{x}\right)=\mathrm{2sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\mathrm{cos}\left(\frac{\mathrm{x}}{\mathrm{2}}\right) \\ $$$$\mathrm{1}\underset{−} {+}\mathrm{sin}\left(\mathrm{x}\right)=\left(\mathrm{cos}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\underset{−} {+}\mathrm{sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)^{\mathrm{2}} \\ $$$$\mathrm{since}\:\mathrm{x}\in\left(\mathrm{0},\frac{\pi}{\mathrm{4}}\right)\Rightarrow\sqrt{\mathrm{1}−\mathrm{sin}\left(\mathrm{x}\right)}=\mid\mathrm{cos}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)−\mathrm{sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\mid=\mathrm{cos}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)−\mathrm{sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right) \\ $$$$\Rightarrow\sqrt{\mathrm{1}+\mathrm{sin}\left(\mathrm{x}\right)}−\sqrt{\mathrm{1}−\mathrm{sin}\left(\mathrm{x}\right)}=\mathrm{2sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right) \\ $$$$\sqrt{\mathrm{1}+\mathrm{sin}\left(\mathrm{x}\right)}+\sqrt{\mathrm{1}−\mathrm{sin}\left(\mathrm{x}\right)}=\left(\mathrm{cos}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)+\mathrm{cos}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)−\mathrm{sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)=\mathrm{2cos}\left(\frac{\mathrm{x}}{\mathrm{2}}\right) \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{2cos}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)}{\mathrm{2sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)}=\mathrm{cot}\left(\frac{\mathrm{x}}{\mathrm{2}}\right) \\ $$$$\mathrm{cot}^{−} \left(\mathrm{f}\left(\mathrm{x}\right)\right)=\mathrm{cot}^{−} \left(\mathrm{cot}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\right)=\frac{\mathrm{x}}{\mathrm{2}} \\ $$

Commented by aliesam last updated on 09/Dec/19