Question

-secx} " align="absmiddle" class="latex-formula"> + $\frac{1}{1+secx}$ = $-2cot^{2} x$

Answer 1

We have the  Trigonometric Identities : secx = 1/cosx; (sinx)^2 + (cosx)^2 = 1;
Then, 1 / (1-secx) = 1 / ( 1 - 1/cosx) = 1 / [(cosx - 1)/cosx] = cosx /
(cosx - 1 ) ;
Similar, 1 / (1+secx) = cosx / (1 + cosx) ;
 cosx / (cosx - 1)  +  cosx / (1 + cosx) = [cosx(1 + cosx) + cosx (cosx - 1)] / [ (cosx - 1)(cox + 1)] =[cosx( 1 + cosx + cosx - 1 )] / [ (cosx - 1)(cox + 1)] =  2(cosx)^2 / [(cosx)^2 - (sinx)^2] =  2(cosx)^2 / (-1) = - 2(cosx)^2;