Question

cosec2A = tan(45+ A) + tan (45- A) / tan (45+A) - tan (45-A)​

Answer 1

show that tan(45+A) + tan(45-A)/ tan(45+A) - tan(45-A)= cosec2A

Numerator = tan(45+A) + tan(45-A)

Denominator = tan(45+A) - tan(45-A)

Tan(A + B) = (TanA + TanB)/(1 - TanATanB)

Tan(A - B) = (TanA - TanB)/(1 + TanATanB)

tan(45+A) = (1 + TanA)/(1 - TanA)

tan(45-A) = (1 - TanA)/(1 + TanA)

Numerator =

(1 + TanA)/(1 - TanA) +  (1 - TanA)/(1 + TanA)

= ((1 + TanA)² + (1 -TanA)²)/(1 - Tan²A)

=( 1 + Tan²A + 2TanA + 1 + Tan²A - 2TanA) /(1 - Tan²A)

= 2(1 + Tan²A) /(1 - Tan²A)

Similarly denominator

(1 + TanA)/(1 - TanA) - (1 - TanA)/(1 + TanA)

= ((1 + TanA)² - (1 -TanA)²)/(1 - Tan²A)

=( 1 + Tan²A + 2TanA - 1 - Tan²A + 2TanA) /(1 - Tan²A)

= 4TanA /(1 - Tan²A)

LHS =

Numerator/ Denominator =  2(1 + Tan²A)/4TanA

1 + Tan²A = Sec²A

= 2Sec²A/4TanA

= 1/2Cos²ATanA

= 1/2CosACosATanA

= 1/2CosASinA

= 1/Sin2A

= Cosec2A

= RHS

QED