Question

How do you prove:
csc^2@cos^2@ = csc^2@ -1

Answer 1

`LHS\\ \\ ={ cosec }^( 2 )\theta { cos }^( 2 )\theta \\ \\ ={ cosec }^( 2 )\theta \left( 1-{ sin }^( 2 )\theta \right)`


`\\ \\ ={ cosec }^( 2 )\theta -{ cosec }^( 2 )\theta { sin }^( 2 )\theta \\ \\ ={ cosec }^( 2 )\theta -\frac { 1 }{ { sin }^( 2 )\theta } \cdot { sin }^( 2 )\theta`


`\\ \\ ={ cosec }^( 2 )\theta -1\\ \\ =RHS`

Answer 2

`cscx=(1)/(sinx);\ sin^2x+cos^2x=1\to cos^2x=1-sin^2x\\----------------------------\\\\csc^2xcos^2x=csc^2x-1\\\\L=csc^2xcos^2x=(1)/(sin^2x)\cdot cos^2x=(cos^2x)/(sin^2x)\\\\R=csc^2x-1=(1)/(sin^2x)-1=(1)/(sin^2x)-(sin^2)/(sin^2x)=(1-sin^2x)/(sin^2x)=(cos^2x)/(sin^2x)\\\\\boxed{L=R}`