Question

How do you prove `1 + sin² x = 1/sec² x`?

Answer 1

The given equation is not true (except for values of `x` equal to integer multiples of `pi`).

Explanation:

`sec(x) = 1/cos(x)`

The maximum value of `abs(cos(x))` is 1

`rArr` the minimum value of `abs(sec(x))` is 1
and
the maximum value of `1/(sec^2(x))` is 1

`sin^2(x)` has a minimum value of 0
`1+sin^2(x)` has a minimum value of 1

The only time `1+sin^2(x) = 1/(sec^2(x))`
is when
`color(white)("XXXX")` `sec^2(x) = 1` and `sin^2(x) = 1`

These conditions are only true for `x=0` and any other value of `x=kpi` (with `kepsilonZZ`).