Question

How do you evaluate `1- \tan ^ { 2} 85^ { \circ } - \csc ^ { 2} 5^ { \circ }`?

Answer 1

`=-2tan^2(85˚)`

Explanation:

Recall that `cos(pi/2 - x) = sinx`, therefore, `sec(85˚) = csc(5˚)` and thus `sec^2(85˚) = csc^2(5˚)`. We now have:

`1 - tan^2(85˚) - sec^2(85˚)`

Now recall that `tan^2x + 1 = sec^2x`, therefore, `1 - sec^2(x) = -tan^2x`.

`-tan^2(85˚) - tan^2(85˚)`

`=-2tan^2(85˚)`

Hopefully this helps!