How do you prove #1/(sinxcosx) - cosx/sinx = tanx#?

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Answer 1 · 2018-06-26T10:32:20

Kindly see a Proof in Explanation.

#1/(sinxcosx)-cosx/sinx#,

#=1/(sinxcosx)-cosx/sinx*cosx/cosx#,

#=(1-cos^2x)/(sinxcosx)#,

#=sin^2x/(sinxcosx)#,

#=sinx/cosx#,

#=tanx#, as desired!

BONUS :

#1/(sinxcosx)-cosx/sinx=2/(2sinxcosx)-cotx=2/(sin2x)-cotx#.

#:. 1/(sinxcosx)-cosx/sinx=tanx rArr 2csc2x-cotx=tanx#.

Answer 2 · 2018-06-26T10:35:32

As proved below.

#"To prove " 1 / (sin x cos x) - cos x / sin x = tan x#

#L H S = 1 / (sin x cos x) - cos x / sin x#

#=> (1 - cos x * cos x) /( sin x cos x), " taking " sin x cos x " as L C M"#

#=>. (1 - cos^2 x) / (sin x cos x), color(crimson)(" identity " sin^2x + cos^2x = 1#

#=> sin^2 x / (sin x cos x)#

#=> sin x / cos x = tan x = R H S#