Question

Find the value of (sec2 theta-1) (1-cosec2 theta)

Answer 1

(sec^2 theta-1)(1-cosec^2theta)
=tan^2 theta×(-cot^2 theta)
{since sec^2 theta-1=tan^2 theta and 1-cosec^2 theta= -cot^2 theta}
=tan^2 theta×(-1)/tan^2 theta
= - 1

Answer 2

Given :

The Trigonometrical equation

( Sec²Ф - 1 )  ( 1 - Cosec²Ф )

To Find :

The value of  ( Sec²Ф - 1 )  ( 1 - Cosec²Ф )

Solution :

As  ( Sec²Ф - 1 )  ( 1 - Cosec²Ф )

The equation can be written as

   ( Sec²Ф - 1 )  [ - (  Cosec²Ф - 1 ) ]

∵ Sec²Ф - 1  = Tan²Ф          ......1

And

Cosec²Ф - Cot²Ф = 1          

Or,   Cosec²Ф - 1 =   Cot²Ф  .........2

So, From given equation, substitute the values

i.e    ( Sec²Ф - 1 )  ( 1 - Cosec²Ф ) =  (  Tan²Ф  ) ( - Cot²Ф )        ..........3

Again

'∵  TanФ =  $\dfrac{1}{Cot\Theta }$

So,   (  $\dfrac{1}{Cot^{2}\Theta }$  ) ( -  Cot²Ф )            ( eq 3 )

    =   $\dfrac{-Cot^{2}\Theta}{Cot^{2}\Theta }$

 ( Sec²Ф - 1 )  ( 1 - Cosec²Ф )     = 1

So, the value of  ( Sec²Ф - 1 )  ( 1 - Cosec²Ф ) = - 1

Hence, The value of  ( Sec²Ф - 1 )  ( 1 - Cosec²Ф ) is - 1   . Answer