Question

Simply (tan^2 x - sec^2 x)(sin^2 x + cos^2 x)

Answer 1

we can use pythagorean identieis

remember that
`sin^2(x)+cos^2(x)=1`

also remember that
`tan(x)=\frac{sin(x)}{cos{x}}` and
`sec(x)=(1)/(cos(x))`

so if we take
`sin^2(x)+cos^2(x)=1` and divide both sides by
`cos^2(x)` we get

`(sin^2(x))/(cos^2(x))+(cos^2(x))/(cos^2(x))=(1)/(cos^2(x))`

`((sin(x))/(cos(x)))^2+1=((1)/(cos(x)))^2`

`tan^2(x)+1=sec^2(x)`

subtracting
`1+sec^2(x)` from both sides

`tan^2(x)-sec^2(x)=-1`

now subsitute into original problem

`(tan^2(x)-sec^2(x))/(sin^2(x)+cos^2(x))=`

`(-1)/(1)=`

`-1`

the answer is -1