Question

Tan^2a + cot^2a + 2=sec^2acosec^2a

Answer 1

To prove ---> tan²A + Cot²A + 2 = Sec²A Cosec²A

Proof---> LHS

= tan²A + Cot²A +2

= ( tanA )² +( CotA )² + 2 tanA ( 1 / tanA )

We know that , 1 / tanA = CotA , applying it here we get,

= ( tanA )² + ( CotA )² + 2 tanA CotA

We have am identity as follows,

( a + b )² = a² + b² + 2ab , we get

= ( tanA + CotA )²

We know that,

tanA = SinA /CosA , CotA = CosA/ SinA , applying it here we get,

= { ( SinA / CosA ) + ( CosA + SinA ) }²

Taking LCM as SinA CosA, we get,

= { ( Sin²A + Cos²A ) / SinA CosA }²

We know that Sin²A + Cos²A = 1 , applying it we get,

= ( 1 / SinA CosA )²

= 1 / Sin²A Cos² A

= ( 1 / Sin²A ) ( 1 / Cos²A )

We know that,

CosecA = 1 / SinA , and SecA = 1 / CosA , applying it , we get,

= Cosec²A Sec²A = RHS