Question

Question #e9930

Answer 1

see explanation.

Explanation:

Utilise the `color(blue)"trigonometric identity"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(sin^2theta+cos^2theta=1)color(white)(2/2)|)))`

From which: `sin^2theta=1-cos^2theta;cos^2theta=1-sin^2theta`

`• 1/(sin^2theta)-(cos^2theta)/(sin^2theta)`

Since both fractions have a common denominator we can subtract the numerators while leaving the denominator.

`=(1-cos^2theta)/(sin^2theta)`

`=cancel(sin^2theta)^1/cancel(sin^2theta)^1larrcolor(red)("from above identity"`

`=1=" right side "rArr" verified"`

`• 1/(cos^2theta)+1/(sin^2theta)`

To obtain a`color(blue)" common denominator".`
multiply the numerator/denominator of `1/(cos^2theta)" by " sin^2theta"`
multiply the numerator/denominator of `1/(sin^2theta)" by " cos^2theta`

`rArr(sin^2theta)/(cos^2thetasin^2theta)+(cos^2theta)/(sin^2thetacos^2theta)`

`=(sin^2theta+cos^2theta)/(cos^2thetasin^2theta)`

`=1/(cos^2thetasin^2theta)larrcolor(red)" from above identity"`

`"Thus left side "=" right side "rArr" verified"`

`• 1/(1+sintheta)+1/(1-sintheta)`

To obtain a `color(blue)"common denominator"`

multiply numerator/denominator of fraction on left by `1-sintheta`

multiply the one on the right by `1+sintheta`

`=(1-sintheta)/((1+sintheta)(1-sintheta))+(1+sintheta)/((1-sintheta)(1+sintheta))`

`=(1cancel(-sintheta)+1cancel(+sintheta))/(1-sin^2theta)`

`=2/cos^2thetalarrcolor(red)" from above identity""`

`"left side "=" right side " rArr" verified"`