Question

prove that secA-1/secA+1=(sinA/1+cosA)=(cotA-cosecA)2

Answer 1

3 answers · Mathematics 

 Best Answer

sec A -1 
------------- 
sec A+1 

Multiplying n dividing by sec A -1, 

(sec A-1)^2 
----------------- 
(sec^2A-1) 

1 
--------- ( sec^2A +1-2secA) ..............................[ sec^2A - 1 = tan^2 A] 
tan^2A 

cot^2a ( tan^2A+2-2secA) ................................ [ 1/tanA = cotA ] 

1+2cot^2A -2 . 1/cosA . cosA/sinA . cosA/sinA..............[simply multiplying] 
........................................... [ cosA/sinA =cotA ; 1/cosA=secA] 
( 1+cot^2A)+cot^2A-2cotAcosecA 

cosec^2a+cot^2A-2cosec^2Acot^2A ........................................... = cosec^2 A] 

(cotA-cosecA)^2..............(Q.E.D)

Answer 2

$\huge{\underline{\sf{Solution:-}}}$

$\implies$sec A - 1 \ sec A + 1 

$\bigstar\sf{Multiplying\:and\:dividing\:by\:sec A - 1}$

$\implies$(sec A - 1)² / (sec²A - 1) 

$\implies$1 × (sec²A + 1 - 2secA) → [sec²A - 1 = tan² A]

$\bigstar\sf{Dividing\:question\:by\:tan^{2}A}$

$\implies$cot²A (tan²A + 2 - 2secA) → [1 / tanA = cotA] 

$\implies$1 + 2cot²A - 2 × 1/cosA × cosA/sinA × cosA/sinA → [simply multiplying] 

$\implies$[cosA / sinA = cotA × 1/cosA = secA] 

$\implies$(1 + cot²A) + cot²A -2cotAcosecA 

$\implies$cosec²A + cot²A - 2cosec²Acot²A = cosec² A

$\implies$(cotA - cosecA)²

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