Prove that:
(i) √secθ-1/secθ+1+√secθ+1/secθ-1=2cosecθ
(ii) √1+sinθ/1-sinθ+√1-sinθ/1+sinθ=2secθ
(iii) √1+cosθ/1-cosθ+√1-cosθ/1+cosθ=2cosecθ
(iv)secθ-1/secθ+1=(sinθ/1+cosθ)²
Answers
√secθ-1/secθ+1+√secθ+1/secθ-1=2cosecθ
Step-by-step explanation:
√secθ-1/secθ+1+√secθ+1/secθ-1=2cosecθ
LHS = √secθ-1/secθ+1+√secθ+1/secθ-1
on rationalizing each term
=> √(secθ-1)²/(sec²θ-1) +√(secθ+1)²/(sec²θ-1)
= √(secθ-1)²/(tan²θ) +√(secθ+1)²/(tan²θ)
= (secθ-1)/tanθ +(secθ+ 1)/tanθ
= 2Secθ/tanθ
= (2/Cosθ)/(Sinθ/Cosθ)
= 2/Sinθ
= 2Cosecθ
= RHS
QED
proved
√secθ-1/secθ+1+√secθ+1/secθ-1=2cosecθ
(ii) √1+sinθ/1-sinθ+√1-sinθ/1+sinθ=2secθ
LHS = √1+sinθ/1-sinθ+√1-sinθ/1+sinθ
on rationalizing each term
=> (1 + sinθ)/Cosθ + (1 - sinθ)/Cosθ
= 2/Cosθ
= 2Secθ
= RHS
(iii) √1+cosθ/1-cosθ+√1-cosθ/1+cosθ=2cosecθ
LHS = √1+cosθ/1-cosθ+√1-cosθ/1+cosθ
on rationalizing each term
(1+cosθ)/Sinθ + (1-cosθ)/Sinθ
= 2/Sinθ
= 2Cosecθ
= RHS
(iv)secθ-1/secθ+1=(sinθ/1+cosθ)²
LHS = secθ-1/secθ+1
(1/Cosθ - 1)/(1/Cosθ + 1)
= (1 - Cosθ)/(1 + Cosθ)
Multiplying and dividing by 1 + Cosθ
= ( 1 - Cos²θ)/(1 + Cosθ)²
=Sin²θ/(1 + Cosθ)²
= (Sinθ/(1 + Cosθ))²
= RHS
QED
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