# If `0 lt= theta lt= 90` , then the value of   `(5 costheta - 4)/(3 - 5 sintheta) - (3 + 5 sintheta)/(4 + 5 costheta)` is a. 0 b. 1 c. 2 d. 4

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`(5costheta-4)/(3-5sintheta)-(3+5sintheta)/(4+5costheta)`

To subtract, express the fractions with same denominator. Since the LCD is `(3-5sintheta)(4+5costheta)` , the fractions become:

`= (5costheta-4)/(3-5sintheta)*(4+5costheta)/(4+5costheta) - (3+5sintheta)/(4+5costheta)*(3-5sintheta)/(3-5sintheta)`

`= ((5costheta-4)(4+5costheta))/((3-5sintheta)(4+5costheta)) - ((3+5sintheta)(3-5sintheta))/((3-5sintheta)(4+5costheta))`

`= (25cos^2theta - 16)/((3-5sintheta)(4+5costheta)) - (9-25sin^2theta)/((3-5sintheta)(4+5costheta)) `

Now that they have same denominator, proceed to subtract.

`=( 25cos^2theta -16 - (9-25sin^2theta))/((3-5sintheta)(4+5costheta))`

`= (25cos^2theta -16-9+25sin^2theta)/((3-5sintheta)(4+5costheta))`

`= (25cos^2theta+25sin^2theta-25)/((3-5sintheta)(4+5costheta))`

`= (25(cos^2theta +sin^2theta) - 25)/((3-5sintheta)(4+5costheta))`

To simplify the numerator further, apply the Pythagorean identity.

`=(25*1 - 25)/((3-5sintheta)(4+5costheta))`

`=(25-25)/((3-5sintheta)(4+5costheta))`

`=0/((3-5sintheta)(4+5costheta))`

`=0`

Hence,  `(5costheta-4)/(3-5sintheta)-(3+5sintheta)/(4+5costheta)= 0 ` .