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

2 Answers
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#

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