Question

Please help!! Prove that secxcscx(tanx+cotx) equals 2+tan^2x+cot^2x

Answer 1

Explanation:

`sec(x)csc(x)(tan(x)+cot(x)) = 2 + tan ^2(x) + cot^2(x)`

First, begin by simplifying the terms of each component listed in the problem:

`sec(x)csc(x)( tan(x) + cot(x)) = ((1)/(cos(x)) )((1)/(sin(x)))((sin(x))/(cos(x)) +(cos(x))/(sin(x)) ))`

Then, begin by distributing
`(1)/(cos(x))` and
`(1)/(sin(x))` into the parentheses:

`((sin(x))/(cos(x)) ((1)/(cos(x)) )((1)/(sin(x))) +(cos(x))/(sin(x)) ((1)/(cos(x)) )((1)/(sin(x))))`

Next, cancel equivalent terms with one another:

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

Further simplify each term:

`sec^2(x) + cos^2(x)`

Using the Pythagorean Identities, allow
`csc^2(x) = 1 + cot^2(x)` and
`sec^2(x) = 1 + tan^2(x)`:

`(1 + tan^2(x)) + (1 + cot^2(x))`

Simplify:

`2 + tan ^2(x) + cot^2(x) = 2 + tan ^2(x) + cot^2(x)`