If A = a cosec θ + bcot θ and B = a cot θ + b cosec θ, then what is the value of $A^2 - B^2 $ ?

$(a^2 - b^2)$

We are given that :-

A = a cosec θ + bcot θ and B = a cot θ + b cosec θ

Now,

A² - B²

= ( a cosec θ + bcot θ )² - ( a cot θ + bcosec θ )²

=  a² cosec² θ + b²cot² θ + 2 ab cosec θ . bcot θ   - a² cot² θ - b²cosec² θ - 2 ab cosec θ . bcot θ

= a² cosec² θ + b²cot² θ  - a² cot² θ - b²cosec² θ

= a² ( cosec² θ - cot² θ ) + b² ( cot² θ - cosec² θ )

{ using , cosec² θ - cot² θ = 1 }

= a² ( 1 ) + b² ( - 1 ) 

= a²  - b²