How can this be simplified to the most basic form? Please include a step-by-step explanation.

#sec^4(x) - tan^4(x)#

1 Answer
martin23239
Mar 1, 2018

# \quad "An answer is:" \qquad \qquad \ sec^4(x) - tan^4(x) = \ 1 + 2 tan^2(x). #

Explanation:

# "We have:" #

# \qquad sec^4(x) - tan^4(x) #

# \qquad \qquad = \ [ sec^2(x) ]^2 - [ tan^2(x) ]^2; #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \ color{blue}{"apply difference of two squares" \quad rArr } #

# \qquad \qquad = \ [ sec^2(x) - tan^2(x) ] cdot [ sec^2(x) + tan^2(x) ]; #

# \qquad \qquad \qquad \qquad \quad color{blue}{"apply twice, trig identity:" \qquad 1 + tan^2x = sec^2x \quad rArr } #

# \qquad \quad = \ [ ( 1 + tan^2(x) ) - tan^2(x) ] cdot [ ( 1 + tan^2(x) ) + tan^2(x) ]; #

# \qquad \qquad = \ [ 1 + tan^2(x) - tan^2(x) ] cdot [ 1 + tan^2(x) + tan^2(x) ]; #

# \qquad \qquad = \ [ 1 + color{red}cancel{ tan^2(x) - tan^2(x) } ] cdot [ 1 + 2 tan^2(x) ]; #

# \qquad \qquad = \ 1 cdot [ 1 + 2 tan^2(x) ]; #

# \qquad \qquad = \ 1 + 2 tan^2(x). #

# "This an answer simplified to a basic form -- always, there may" #
# "be other basic forms to which it can be simplified, as well; but" #
# "this is one." #

# "Summary:" #

# \qquad \qquad \qquad \qquad \qquad \qquad sec^4(x) - tan^4(x) = \ 1 + 2 tan^2(x). #

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