Question 5 (x)
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(1+tan2A1+cot2A)=(1−tanA1−cotA)2=tan2A
(i) 1+tan2A1+cot2A)=1+tan2A1+1tan2A)=1+tan2A[(1+tan2A)tan2A]=tan2A
(ii) (1−tanA1−cotA)2=[1−sinAcosA1−cosAsinA]2=(sin2A)×(cosA−sinA)2(cos2A)×(sinA−cosA)2=(sin2A)(cos2A)×(−1)2=sin2Acos2A=tan2A
∴ (1+tan2A1+cot2A)=(1−tanA1−cotA)2=tan2A.