Answer:
1. [tex]sec(x)-1[/tex]
2. [tex]2 sec^2(x)[/tex]
3. [tex]sec(x)-1[/tex]
Step-by-step explanation:
1. [tex]tan(x)[csc(x) - cot(x)][/tex]
expand the brackets:
[tex]\implies tan(x)csc(x)-tan(x)cot(x)[/tex]
Substitute the following equivalents:
[tex]tan(x)=\dfrac{sin(x)}{cos(x)}[/tex], [tex]csc(x)=\dfrac{1}{sin(x)}[/tex], [tex]cot(x)=\dfrac{1}{tan(x)}[/tex]
[tex]\implies \dfrac{sin(x)}{cos(x)} \cdot \dfrac{1}{sin(x)}-\dfrac{tan(x)}{1} \cdot \dfrac{1}{tan(x)}[/tex]
Cancel the common factors:
[tex]\implies \dfrac{1}{cos(x)}-1[/tex]
[tex]\implies sec(x)-1[/tex]
2. [tex]\dfrac{1}{sin(x)+1}-\dfrac{1}{sin(x)-1}[/tex]
[tex]\implies \dfrac{(sin(x)-1)-(sin(x)+1)}{(sin(x)+1)(sin(x)-1)}[/tex]
[tex]\implies \dfrac{-2}{sin^2(x)-1}[/tex]
Using identity [tex]sin^2(x)+cos^2(x)=1[/tex]
[tex]\implies \dfrac{-2}{-cos^2(x)}[/tex]
[tex]\implies \dfrac{2}{cos^2(x)}[/tex]
[tex]\implies 2 sec^2(x)[/tex]
3. [tex]\dfrac{tan^2(x)}{sec(x)+1}[/tex]
Using identity [tex]1 + tan^2(x)=sec^2(x)[/tex]:
[tex]\implies \dfrac{sec^2(x)-1}{sec(x)+1}[/tex]
[tex]\implies \dfrac{(sec(x)+1)(sec(x)-1)}{sec(x)+1}[/tex]
Cancel common factor [tex]sec(x)+1[/tex]:
[tex]\implies sec(x)-1[/tex]
