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Question

Question 5 (iii)
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(iii) tanθ(1cotθ)+cotθ(1tanθ)=1+secθcosecθ
[Hint : Write the expression in terms of sinθ and cosθ]

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Solution

(iii)tanθ(1cotθ)+cotθ(1tanθ)=1+secθcosecθ
L.H.S.=tanθ(1cotθ)+cotθ(1tanθ)=[(sinθcosθ)1(cosθsinθ)]+[(cosθsinθ)1(sinθcosθ)]=[(sinθcosθ)(sinθcosθ)sinθ]+[(cosθsinθ)(cosθsinθ)cosθ]
=[sin2θcosθ(sinθcosθ)]+[cos2θsinθ(cosθsinθ)]=[sin2θcosθ(sinθcosθ)][cos2θsinθ(sinθcosθ)]=1(sinθcosθ)[(sin2θcosθ)(cos2θsinθ)]=1(sinθcosθ)×[(sin3θcos3θ)sinθcosθ]
=[(sinθcosθ)(sin2θ+cos2θ+sinθcosθ)][(sinθcosθ)sinθcosθ]=(1+sinθcosθ)sinθcosθ=1sinθcosθ+1=1+secθcosecθ=R.H.S.

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