Question

prove the following identity: sec x csc x(tan x + cot x) = 2+tan^2 x + cot^2 x please provide a proof in some shape form or fashion :/

Answer 1

Answer:

Step-by-step explanation:

Hello,

Is this equality true ?

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

1. let 's estimate the left part of the equation

$sec(x)csc(x)(tan(x) + cot(x)) =\dfrac{1}{cos(x)sin(x)}*(\dfrac{sin(x)}{cos(x)}+\dfrac{cos(x)}{sin(x)})\\\\=\dfrac{1}{cos(x)sin(x)}*(\dfrac{sin^2(x)+cos^2(x)}{sin(x)cos(x)})\\\\=\dfrac{1}{cos(x)sin(x)}*(\dfrac{1}{sin(x)cos(x)})\\\\\\=\dfrac{1}{cos^2(x)sin^2(x)}$

1. let 's estimate the right part of the equation

$2+tan^2(x) + cot^2(x)=2+\dfrac{sin^2(x)}{cos^2(x)}+\dfrac{cos^2(x)}{sin^2(x)}\\\\=\dfrac{2cos^2(x)sin^2(x)+cos^4(x)+sin^4(x)}{cos^2(x)sin^2(x)}\\\\=\dfrac{(cos^2(x)+sin^2(x))^2}{cos^2(x)sin^2(x)}\\\\=\dfrac{1^2}{cos^2(x)sin^2(x)}\\\\=\dfrac{1}{cos^2(x)sin^2(x)}$

This is the same expression

So

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

hope this helps