How do you use the fundamental trigonometric identities to determine the simplified form of the expression?

"The fundamental trigonometric identities" are the basic identities:

•The reciprocal identities
•The pythagorean identities
•The quotient identities

They are all shown in the following image:

When it comes down to simplifying with these identities, we must use combinations of these identities to reduce a much more complex expression to its simplest form.

Here are a few examples I have prepared:

a) Simplify: # tanx/cscx xx secx#

Apply the quotient identity #tantheta = sintheta/costheta# and the reciprocal identities #csctheta = 1/sintheta# and #sectheta = 1/costheta#.

#=(sinx/cosx)/(1/sinx) xx 1/cosx#

#=sinx/cosx xx sinx/1 xx 1/cosx#

#=sin^2x/cos^2x#

Reapplying the quotient identity, in reverse form:

#=tan^2x#

b) Simplify: #(cscbeta - sin beta)/cscbeta#

Apply the reciprocal identity #cscbeta = 1/sinbeta#:

#=(1/sinbeta - sin beta)/(1/sinbeta)#

Put the denominator on a common denominator:

#=(1/sinbeta - sin^2beta/sinbeta)/(1/sinbeta)#

Rearrange the pythagorean identity #cos^2theta + sin^2theta = 1#, solving for #cos^2theta#:

#cos^2theta = 1 - sin^2theta#

#=(cos^2beta/sinbeta)/(1/sinbeta)#

#=cos^2beta/sinbeta xx sin beta/1#

#=cos^2beta#

c) Simplify: #sinx/cosx + cosx/(1 + sinx)#:

Once again, put on a common denominator:

#=(sinx(1 + sinx))/(cosx(1 + sinx)) + (cosx(cosx))/(cosx(1 + sinx))#

Multiply out:

#=(sinx + sin^2x + cos^2x)/(cosx(1 + sinx))#

Applying the pythagorean identity #cos^2x + sin^2x = 1#:

#=(sinx + 1)/(cosx(1 + sinx))#

Cancelling out the #sinx + 1# since it appears both in the numerator and in the denominator.

#=cancel(sinx + 1)/(cosx(cancel(sinx + 1))#

#=1/cosx#

Applying the reciprocal identity #1/costheta = sectheta#

#=secx#

Finally, on a last note, I know that here in Canada, British Columbia more specifically, these identities are given on a formula sheet, but I don't know what it's like elsewhere. In any event, many students, me included, memorize these identities because they're that important to mathematics. I would highly recommend memorization.

Practice exercises:

Simplify the following expressions:

a) #cosalpha + tan alphasinalpha#

b) #cscx/sinx - cotx/tanx#

c) #sin^4theta - cos^4theta#

d) #(tan beta + cot beta)/csc^2beta#

Hopefully this helps, and good luck!