Verify each trigonometric equation by substituting identities to match the right hand side of the equation to the left hand side of the equation.
cot x sec^4 x = cot x + 2 tan x + tan^3 x
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Step-by-step explanation:
cot x sec^4 x = cot x + 2 tan x + tan^3 x
Taking left hand side,
= cotx + 2 tanx + tan^3×
= ( cotx / sinx ) + ( 2sinx / cosx ) + ( sin^3× / cos^3× )
Taking L.C.M;
= (cos^4× + 2sin^2× cos^2× + sin^4×) / sinx cos^3×
Using formula a^2 + b^2 + 2ab = (a + b)^2
= (cos^2× + sin^2×)^2 / sinx cos^3×
As cos^2× + sin^2× = 1
= 1 / sinx cos^3×
= cosx / sinx cos^4×
= (cos^4× + 2sin^2× cos^2× + sin^4×) / sinx cos^3×
Using formula a^2 + b^2 + 2ab = (a + b)^2
= (cos^2× + sin^2×)^2 / sinx cos^3×
As cos^2× + sin^2× = 1
= 1 / sinx cos^3
= cosx / sinx cos^4×
= ( cos × / sinx )* ( 1/cos^4× )
cotx sec^4×
Hence right hand and left side are equal.
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