Question

How do you prove `1/sec^2 x + 1/csc^2 x =1`?

Answer 1

`1/csc^(2⁡)x +1/sec^(2)x =1`

`1/(1/sin^(2⁡)x )+1/(1/cos^(2⁡)x )=1`

`sin^2⁡x+cos^2⁡x=1`

`1=1`

Explanation:

1. Use reciprocal identity to transform `1/csc^2 x` and `1/sec^2 x` to `1/[1/ (sin^2 x)]` and `1/[1/(cos^2 x)]` respectively.

2. Reciprocate `sin^2 x` and `1/(cos^2 x)` you will get `sin^2x` and `cos^2 x`, respectively.

3. `sin^2⁡x+cos^2⁡x` is equal to `1` by the Pythagorean Fundamental Identity.