Simplify: (sin(x-pi))/(cos (pi+x)) - (cos (pi/2-x))/(sin(-pi-x))

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to simplify: (sin(x-pi))/(cos (pi+x)) - (cos (pi/2-x))/(sin(-pi-x))

We use the relation sin(a+b) = sin a*cos b + cos a*sin b and cos( a +b) = cos a*cos b - sin a * sin b.

(sin(x-pi))/(cos (pi+x)) - (cos (pi/2-x))/(sin(-pi-x))

=>[(sin x * cos pi - cos x*sin pi) / (cos pi* cos x - sin pi * sin x)] - [(cos pi/2 *cos x + sin pi/2 *sin x) / (-sin pi*cos x - cos pi*sin x)]

sin pi = 0, cos pi = -1, sin pi/2 = 1 and cos pi/2 = 0

=> [(- sin x) / (-cos x)] - [(sin x) / (+sin x)]

=> tan x - 1

The required simplified result is tan x - 1.

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lochana2500 | Student, Undergraduate | (Level 1) Valedictorian

Posted on

sin(x-π)/cos (π+x) ⇒ ❶ 

= -sin(π-x)/-cosx

= -sinx/-cosx

= tanx

cos (π/2-x)/sin(-π-x) ⇒ ❷

= sinx/-sin(π+x)

= sinx/sinx

=1

Hence ❶ - ❷ = tanx-1

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