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 $ ?
Answer & explanation
Correct answer: option 3
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