How do you simplify the following?

#(1sin^2 x)/(sin x + 1)#

#(tan x)(1sin^2 x)#

#(1sin^2 x)/(sin x + 1)# 
#(tan x)(1sin^2 x)#
1 Answer
Mar 26, 2016

#(1sin^2 x)/(sin x + 1) = 1  sin x# when#x != (3pi)/2 + 2kpi# 
#(tan x)(1  sin^2 x) = 1/2 sin 2x# when#x != kpi#
Explanation:
Example 1.
Use the difference of squares identity:
#a^2b^2 = (ab)(a+b)#
with
#(1sin^2 x)/(sin x + 1) = ((1sin x)color(red)(cancel(color(black)((1+sin x)))))/color(red)(cancel(color(black)((1+sin x)))) = 1  sin x#
with exclusion
Example 2.
Use the following:
#sin^2 x + cos^2 x = 1# in the form#1  sin^2 x = cos^2 x#
#tan x = (sin x)/(cos x)#
#sin 2x = 2 sin x cos x#
as follows:
#(tan x)(1  sin^2 x) =(sin x)/(cos x)*cos^2 x = sin x cos x = 1/2 sin 2x#
with exclusion