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

2 Answers
Hriman
Mar 21, 2018

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

Steve M
Mar 21, 2018

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

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