how to verify cot^2x+cos-csc x cot x=(cos x)/(1+sec x)?

joseph

how to verify this indentity

g-boy-ss

cot ² x + cos x - csc x cot x = cos x / ( 1 + sec x )

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

..... ..... ..... ..... .... ..... ..... = cos x / {( cos x + 1 ) / cos x }

( cos ² x + cos x sin ² x - cos x ) / sin ² x = cos ² x / ( cos x + 1 )

cos x ( cos x + sin ² x - 1 ) / sin ² x = cos ² x ( 1 - cos x ) / ( 1 + cos x ) ( 1 - cos x )

cos x ( cos x - cox ² x ) / sin ² x = cos ² x ( 1 - cos x ) / ( 1 - cos ² x )

cos ² x ( 1 - cox x ) / sin ² x = cos ² x ( 1 - cos x ) / sin ² x ) ◄ ← ← ← proven !!!

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mohanrao-d

cot^2(x) + cos(x) - csc(x) cot( x)

= { cos^2(x) / sin^2(x) } + cos x - { cos x / sin^2(x) }

= { cos^2(x) + cos x sin^2(x) - cos x ) / sin^2(x)

= cos x [ cos x + sin^2(x) - 1 ] / sin^2(x)

= cos x [ cos x - cos^2(x) ] / sin^2(x)

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

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

= cos^2(x) ( 1 - cos x ) / (1 + cos x)(1 - cos x )

= cos^2(x) / (1 + cos x )

= cos x / sec x (1 + cos x )

= cos x / (1 + sec x )

=

moe

cot^2x+cos-csc x cot x=(cos x)/(1+sec x)

LHS: cos^2x/sin^2x + cosx -1/sinx*(cosx/sinx)

=[cos^2x + cosx(sin^2x-1)]/sin^2x

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

RHS: cos^x/(cosx + 1)

cos^2x(1-cosx)/sin^2x = cos^x/(cosx + 1) :divide both sides by cos^2x

(1-cosx)(1+cosx) = sin^x :a^2-b^2 = (a+b)(a-b)

1-cos^x = sin^x

LHS=RHS

fazaldin-a

cot^2x+cos-csc x cot x=(cos x)/(1+sec x)?

l.h.s. = cot^2x+cos-csc x cot x = cos^2x/sin^2x +cosx -cosx/sin^2x =

= (cos^2x +cosx.sin^2x -cosx)/sin^2x = cosx(cosx +sin^2x-1)/sin^2x =

= cosx(cosx - cos^2x)/sin^2x = (cos x)/(1+sec x) = R.H.S. >=============< Q.E.D.