Prove that by identities
(i) (1-cos𝜃) (1+ cos𝜃) ( 1+cot2ፀ)=1
solution:-
LHS= (1-cos𝜃) (1-cos𝜃)( 1+cot2ፀ)
=( 1- cos2ፀ) ( 1+cot2ፀ)
=(sin2ፀ) (cosec2ፀ)
=1 =RHS proved...
(ii) {(1+sinፀ)2+ (1 - sinፀ)2}/2cos2ፀ = (1+sin2ፀ)/(1 -sin2ፀ)
solution:-
LHS= {(1+sinፀ)2+ (1 - sinፀ)2}/2cos2ፀ
={1 +2sinፀ+ sin2ፀ + 1 - 2sinፀ + sin2ፀ}/2cos2ፀ
=1+ sin2ፀ +1+ sin2ፀ /2cos2ፀ
=2(1+ sin2ፀ)/2(1 - sin2ፀ)
=(1+sin2ፀ)/(1 -sin2ፀ) =RHS proved
(iii) cos2ፀ(1-cos𝜃)/sin2ፀ (1 - sinፀ) = (1+sinፀ)/(1+ cos𝜃)
solution:-
LHS= cos2ፀ(1-cos𝜃)/sin2ፀ (1 - sinፀ)
=(1 -sin2ፀ)(1-cos𝜃)/(1- cos2ፀ ) (1 - sinፀ)
= (1 - sinፀ) (1 + sinፀ) (1-cos𝜃)/(1-cos𝜃) (1+cos𝜃) (1 - sinፀ)
= (1 + sinፀ)/ (1+cos𝜃) = RHS proved
(iv) (sinፀ - cosፀ)2= 1 - 2 sinፀ cosፀ
solution:-
LHS= (sinፀ - cosፀ)2
= sin2ፀ - 2 sinፀ cosፀ + cos2ፀ ( we know that sin2ፀ + cos2ፀ = 1 )
= 1 - 2 sinፀ cosፀ = RHS proved
(v) (sinፀ + cosፀ)2+ (sinፀ - cosፀ)2= 2
solution:-
LHS =(sinፀ + cosፀ)2+ (sinፀ - cosፀ)2
= sin2ፀ + 2 sinፀ cosፀ + cos2ፀ + sin2ፀ - 2 sinፀ cosፀ + cos2ፀ ( we know that sin2ፀ + cos2ፀ = 1 )
= 1 + 1 = 2 = RHS proved...
