Tan^3x/1+tan^2x +cot^3x/1+cot^2x = secx cosecx - 2sincosx
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tan^3x / (1 + tan^2x ) +
cot^3x / ( 1 + cot^2x )
=tan^3x / sec^2x + cot^3x /cosec^2x
=( sin^3x / cos^3x ) / 1 / cos^2x +
( cos^3x/sin^3x ) / 1 / sin^2x
= (sin^3x / cos^3x ) × ( cos^2x / 1 ) +
( cos^3x / sin^3x ) ×( sin^2 x / 1 )
= sin^3x / cosx + cos^3 x / sinx
= sin^4x + cos^4 x / cosx sinx
=( sin^2x)^2 + (cos^2x)^2 +
2sin^2x cos^2x - 2sin^2x cos^2x / cosx sinx
=( sin^2x + cos^2x )^2-2sin^2x cos^2x /cosx sinx
= (1)^2 - 2sin^2x cos^2x / cosx sinx
= 1 / sinx cosx - 2sin^2x cos^2x/ cosx sinx
= secx cosecx - 2sinx cos x
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★★★★★★hope it helps ★★★★★★★★★★★★★
tan^3x / (1 + tan^2x ) +
cot^3x / ( 1 + cot^2x )
=tan^3x / sec^2x + cot^3x /cosec^2x
=( sin^3x / cos^3x ) / 1 / cos^2x +
( cos^3x/sin^3x ) / 1 / sin^2x
= (sin^3x / cos^3x ) × ( cos^2x / 1 ) +
( cos^3x / sin^3x ) ×( sin^2 x / 1 )
= sin^3x / cosx + cos^3 x / sinx
= sin^4x + cos^4 x / cosx sinx
=( sin^2x)^2 + (cos^2x)^2 +
2sin^2x cos^2x - 2sin^2x cos^2x / cosx sinx
=( sin^2x + cos^2x )^2-2sin^2x cos^2x /cosx sinx
= (1)^2 - 2sin^2x cos^2x / cosx sinx
= 1 / sinx cosx - 2sin^2x cos^2x/ cosx sinx
= secx cosecx - 2sinx cos x
R.H.S★★★★★★★★★★★★★★★★★★★★★★★★
★★★★★★hope it helps ★★★★★★★★★★★★★
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