How do you verify the identity (csc x - sin x)(sec x - cos x)(tan x + cot x) = 0(cscx−sinx)(secx−cosx)(tanx+cotx)=0?

2 Answers
Gió
Apr 23, 2015

I am not sure that is equal to zero....I got 11!
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Have a look.

Alan P.
Apr 23, 2015

WARNING: This may not be the simplest way!
(csc - sin)(sec - cos)(tan+cot)(csc−sin)(sec−cos)(tan+cot)
Let ss represent sin(x)sin(x)
cc represent cos(x)cos(x)
and tt represent tan(x)tan(x)

(csc - sin)(sec - cos)(tan+cot)(csc−sin)(sec−cos)(tan+cot)
becomes
(1/s - s)(1/c-c)(t+1/t)(1s−s)(1c−c)(t+1t)

=(1/(sc)-t - 1/t+sc)*(t+1/t)=(1sc−t−1t+sc)⋅(t+1t)

((xx,|1/(sc),,-t,,-1/t,,+sc),( - , - , - , - , - , - , - , - - ),(1/t,|1/(sct),,-1,,-1/t^2,,+(sc)/t),(+t,|t/(sc),,-t^2,,-1,,sct) )

Replacing t with s/c

we get
1/(s^2) - 1 -c^2/s^2 +c^2 +1/c^2 -s^2/c^2 -1 +s^2

=(1-s^2)/(c^2) -2 +(c^2+s^2) +(1-c^2)/s^2

= 1

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