How can I simplify this trigonometry expression? Thanks!

Whenever I see two fractions being added or subtracted with different denominators, I automatically think we need a common denominator.

#1/(1+sintheta)+1/(1-sintheta)=#
#(1(1-sintheta))/((1+sintheta)(1-sintheta))+(1(1+sintheta))/((1-sintheta)(1+sintheta))=#
#2/(1-sin^2theta)#
Now that we successfully added the two fractions, we will apply a Pythagorean identity to simplify:
#sin^2theta+cos^2theta=1#
Therefore:
#cos^2theta=1-sin^2theta#
Let's substitute #cos^2theta# in our denominator:
#2/(cos^2theta)#
Now apply the reciprocal identity:
#Sectheta=1/costheta#
Final answer: #2sec^2theta#

#1/(1 + sin theta) + 1/(1 - sin theta)#

multiplying each term by one in the form of a ratio of an appropriately chosen numerator and denominator:

#= (1 - sin theta)/((1 + sin theta)(1 - sin theta)) + (1 + sin theta)/((1 - sin theta)(1 + sin theta))#

using the corollary of the difference of two squares for the terms in the denominators:

#= (1 - sin theta)/(1 - sin^2theta) + (1 + sin theta)/(1 - sin^2theta)#

summing terms with the same denominator

#(1 - sin theta + 1 + sin theta)/(1 - sin^2theta)#

using a rearrangement of the identity #cos^2 theta + sin^2 theta = 1#:

#= 2 / cos^2theta#

noting #sec^2 theta = 1/(cos^2 theta)#:

# = 2 sec^2 theta #