Question

(sec(x) sin^2(x)) / (1 + sec(x)) simplifyy pls

Answer 1

`1-\cos \left(x\right)`

Explanation:

`(\sec \left(x\right)\sin ^2\left(x\right))/(1+\sec \left(x\right))`

Use the identity:
`\sec \left(x\right)=(1)/(\cos \left(x\right))`

`((1)/(\cos \left(x\right))\sin ^2\left(x\right))/(1+(1)/(\cos \left(x\right)))`

`((1)/(\cos \left(x\right))\sin ^2\left(x\right))/((\cos \left(x\right)+1)/(\cos \left(x\right)))`

`((\sin ^2\left(x\right))/(\cos \left(x\right)))/((\cos \left(x\right)+1)/(\cos \left(x\right)))`

Divide them using
`((a)/(b))/((c)/(d))=(a* d)/(b* c)`

`(\sin ^2\left(x\right)\cos \left(x\right))/(\cos \left(x\right)\left(\cos \left(x\right)+1\right))`

Cancel cos(x):

`(\sin ^2\left(x\right))/(\cos \left(x\right)+1)`

Use the identity:
`\sin ^2\left(x\right)=1-\cos ^2\left(x\right)`

`(1-\cos ^2\left(x\right))/(1+\cos \left(x\right))`

`(-\left(\cos ^2\left(x\right)-1\right))/(1+cos(x))`

Use the difference of squares formula:

`-(\left(\cos \left(x\right)+1\right)\left(\cos \left(x\right)-1\right))/(1+\cos \left(x\right))`

Cancel out 1+cos(x):

`-\left(\cos \left(x\right)-1\right)`

Remove outer bracket:

`-\cos \left(x\right)+1`

`1-cos(x)`