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!