How do you prove $Sec(x) - cos(x) = sin(x) * tan(x)$ ?

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Answer 1 · 2016-05-30T21:10:30

Recall that $sectheta = 1/costheta$ and that $tantheta = sintheta/costheta$

$1/cosx - cosx = sinx xx sinx/cosx$

Now put the left side on a common denominator:

$1/cosx - (cosx xx cosx)/cosx = sinx xx sinx/cosx$

Simplify:

$(1 - cos^2x)/cosx = sin^2x/cosx$

Now use the modified pythagorean identity $sin^2x + cos^2x = 1$ : $sin^2theta = 1 - cos^2theta$ .

$(sin^2x)/cosx = sin^2x/cosx ->$ Identity proved!!

Practice exercises:

Prove the following identities, using the quotient, pythagorean and reciprocal identities.

a) $1/tanx + tanx = 1/(sinxcosx)$

b) $cos^2x = (cscxcosx)/(tanx + cotx)$

Hopefully this helps, and good luck!

How do you prove Sec(x) - cos(x) = sin(x) * tan(x)Sec(x) - cos(x) = sin(x) * tan(x)?