Question

Prove the following trigonometric identities. $\frac{cotA-cosA}{cotA+cosA}=\frac{cosecA-1}{cosecA+1}$

Answer 1

Answer with Step-by-step explanation:

Given :  

(cotA − cosA)/(cotA + cosA) = (cosecA−1)/(cosecA + 1)

 

LHS : (cotA − cosA)/(cotA + cosA)

=[(cosA/sinA) −cosA] / [(cosA/sinA) + cosA]

[By using an identity, cotθ = cosθ/sinθ ]

=[(cosA × 1/sinA) −cosA] / [(cosA × 1/sinA) + cosA]

= (cosA × cosecA−cosA) / (cosA × cosecA + cosA)  

[By using the identity, 1/sinθ = cosecθ]

= cosA(cosecA − 1) / cosA(cosecA + 1)

= (cosecA − 1)(cosecA +1 )

(cotA − cosA)/(cotA + cosA) = (cosecA−1)/(cosecA + 1)

L.H.S = R.H.S  

Hence Proved..

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Answer 2

Given that :

» (cotA − cosA)/(cotA + cosA) = (cosecA−1)/(cosecA + 1)

Here,

LHS : (cotA − cosA)/(cotA + cosA)

» {(cosA/sinA) −cosA} / {(cosA/sinA) + cosA}

Note: Using identity - cotθ = cosθ/sinθ

» {(cosA × 1/sinA) −cosA} / {(cosA × 1/sinA) + cosA}

» (cosA × cosecA−cosA) / (cosA × cosecA + cosA)  

Note: Using identity - 1/sinθ = cosecθ

» cosA(cosecA − 1) / cosA(cosecA + 1)

» (cosecA − 1)(cosecA +1 )

(cotA − cosA)/(cotA + cosA)

» (cosecA−1)/(cosecA + 1)

∴ L.H.S = R.H.S  ___[Verified]