1 simplify to an integer sec2t1tan2t 2 simplify to an intege

1) simplify to an integer (sec^(2)t-1)/(tan^(2)t)

2) simplify to an integer: (csc^(4)\\theta -cot^(4)\\theta )/(csc^(2)\\theta +cot^(2)\\theta )

3) prove sin^2t-cos^2t=(1-cot^2t)/(1+cot^2t)

4) prove (1+tan s)/(1-tan s)=(sec^2 s+2tan s)/(2-sec^2 s)

Solution

1) (sec^(2)t-1)/(tan^(2)t) = (tan^(2)t)/(tan^(2)t) = 1

2) (csc^(4)\\theta -cot^(4)\\theta )/(csc^(2)\\theta +cot^(2)\\theta )

= (csc^(2)\\theta - cot^(2)\\theta ) = 1

3) LHS= sin^2t-cos^2t = - cos2t

RHS = (1-cot^2t)/(1+cot^2t) = (tan^2t - 1)/(tan^2t + 1) = (sin^2t-cos^2t)/(sin^2t + cos^2t) = -cos2x

LHS = RHS

4) LHS = (1+tan s)/(1-tan s)

RHS = (sec^2 s+2tan s)/(2-sec^2 s) = (1 + tan^2 s + 2tan s)/(2 - 1 - tan^2 s)

= (1 + tans)^2/(1 - tan^2s) = (1 + tans)/(1 - tans) = LHS