How do you simplify #(1+sin^2x)^2 - (sin^2x-1)^2?#

2 Answers
Hriman
Mar 15, 2018

#4sin^2x#

Explanation:

#(1+sin^2x)^2-(sin^2x-1)^2#
#1+2sin^2x+sin^4x -(sin^4x-2sin^2x+1)#
#cancel(1)+2sin^2xcancel(+sin^4x)cancel( -sin^4x)+2sin^2xcancel(-1)=#
#4sin^2x#

Steve M
Mar 16, 2018

# (1+sin^2x)^2 - (sin^2x-1) -= 4sin^2x#

Explanation:

The expression is the difference of two squares so we can use:

# a^2-b^2 -= (a+b)(a-b) #

So we can write:

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

# " " -= {(1+sin^2x)+(sin^2x-1)}{(1+sin^2x)-(sin^2x-1)}#

# " " = (1+sin^2x+sin^2x-1)(1+sin^2x-sin^2x+1)#

# " " = (2sin^2x)(2)#

# " " = 4sin^2x#

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