Question

Verify the identity `sin x cos x(tan x + cot x) = 1`?

Answer 1

Verified below

Explanation:

Using the identities:
`tanx=sinx/cosx`

`cotx=cosx/sinx`

`sin^2x+cos^2x=1`

Start:
`sin x cos x(tan x + cot x) = 1`

`sin x cos xtan x + sin x cosxcot x = 1`

`sin x cancel(cos x)*sinx/cancel(cos x) + cancel(sin x) cosx*cos x/cancel(sin x) = 1`

`sin^2x+cos^2x=1`

`1=1`

Answer 2

We seek to prove that:

`sin x cos x(tan x + cot x) -= 1`

Consider the LHS:

`LHS -= sin x cos x(tan x + cot x)`
`\ \ \ \ \ \ \ \ = sin x cos x(sinx/cosx + cosx/sinx)`
`\ \ \ \ \ \ \ \ = sin x cos x((sinxsinx + cosxcosx)/(sinxcosx))`
`\ \ \ \ \ \ \ \ = sin x cos x((sin^2x + cos^2x)/(sinxcosx))`
`\ \ \ \ \ \ \ \ = sin^2x + cos^2x`
`\ \ \ \ \ \ \ \ -= 1 \ \ \` QED