Prove sec x - tan x sin x = (1/sec x) ?

2 Answers
Sonnhard
Jul 17, 2018

See my proof below

Explanation:

We will simplify the left-hand side of your equation:
#sec(x)-tan(x)*sec(x)=#

#1/cos(x)-sin^2(x)/cos(x)=(1-sin^2(x))/cos(x)#

(since #tan(x)*sin(x)=sin(x)/cos(x)*sin(x)=sin^2(x)/cos(x)#)

further

#(1-sin^2(x))/cos(x)=cos^2(x)/cos(x)=cos(x)=1/sec(x)#

(since #1-sin^2(x)=cos^2(x)#)

Jim G.
Jul 17, 2018

#"see explanation"#

Explanation:

#"using the "color(blue)"trigonometric identities"#

#•color(white)(x)cosx=1/secxhArrsecx=1/cosx#

#•color(white)(x)tanx=sinx/cosx#

#•color(white)(x)cos^2x+sin^2x=1#

#"consider the left side"#

#1/cosx-sinx/cosx xxsinx#

#=1/cosx-sin^2x/cosx#

#=(1-sin^2x)/cosx#

#=cos^2x/cosx#

#=cosx=1/secx=" right side "rArr" verified"#

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