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Evaluate the following limit : limx→π4[tan2x-cot2xsecx-cosecx] - Mathematics and Statistics

Sum

Evaluate the following limit :

`lim_(x -> pi/4) [(tan^2x - cot^2x)/(secx - "cosec"x)]`

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Solution

`lim_(x -> pi/4) (tan^2x - cot^2x)/(secx - "cosec"x)`

= `lim_(x -> pi/4) ((sec^2 x - 1) - ("cosec"^2x - 1))/(secx - "cosec"x)`

= `lim_(x -> pi/4) (sec^2 x - "cosec"^2x)/(secx - "cosec"x)`

= `lim_(x -> pi/4) ((secx - "cosec" x)(secx + "cosec"x))/((secx - "cosec"x))`

= `lim_(x -> pi/4) (sec x + "cosec"x)   ...[(because x -> pi/4","  therefore secx -> sqrt(2) and "cosec"x -> sqrt(2)),(therefore secx - "cosec"x -> 0 therefore secx - "cosec"x ≠ 0)]`

= `lim_(x -> pi/4) (secx) + lim_(x -> pi/4) ("cosec"x)`

= `sec  pi/4 + "cosec" pi/4`

= `sqrt(2) + sqrt(2)`

= `2sqrt(2)`

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