\(\frac{{\sqrt {\ cosec x - 1} }}{{\sqrt {\ cosec x + 1} }}\) is equa

1.	\(\frac{{\sqrt {\ cosec x - 1} }}{{\sqrt {\ cosec x + 1} }}\) is equal to:1. \(\tan x - \sec x\)2. \(\sec x.\tan x\)3. \(\tan x + \sec x\)4. \(\sec x - \tan x\)
Answer» Correct Answer - Option 4 : \(\sec x - \tan x\) Given: √(cosecx – 1)/√(cosecx + 1) Trigonometry identities used: sin²x + cos²x = 1 Formula used: a² – b² = (a + b) (a – b) Calculation: √(cosecx – 1)/√(cosecx + 1) (Here cosecx = 1/sinx √[(1/sinx) – 1]/√[(1/sinx) + 1] = √(1 – sinx)/√(1 + sinx) Rationalization = [√(1 – sinx)/√(1 + sinx)] × [√(1 – sinx)/√(1 – sinx)] = √(1 – sinx)²/√(1 + sinx) (1 – sinx) Here a² – b² = (a + b) (a – b) = (1 – sinx)/√(1 – sin²x) Here 1 – sin²x = cos²x = (1 – sinx)/√cos²x = (1 – sinx)/cosx = (1/cosx) – (sinx/cosx) Here 1/cosx = secx, sinx/cosx = tanx = secx – tanx

\(\frac{{\sqrt {\ cosec x - 1} }}{{\sqrt {\ cosec x + 1} }}\) is equal to:1. \(\tan x - \sec x\)2. \(\sec x.\tan x\)3. \(\tan x + \sec x\)4. \(\sec x - \tan x\)

Answer» Correct Answer - Option 4 : \(\sec x - \tan x\)

Given:

√(cosecx – 1)/√(cosecx + 1)

Trigonometry identities used:

sin²x + cos²x = 1

Formula used:

a² – b² = (a + b) (a – b)

Calculation:

√(cosecx – 1)/√(cosecx + 1)

(Here cosecx = 1/sinx

√[(1/sinx) – 1]/√[(1/sinx) + 1]

= √(1 – sinx)/√(1 + sinx)

Rationalization

= [√(1 – sinx)/√(1 + sinx)] × [√(1 – sinx)/√(1 – sinx)]

= √(1 – sinx)²/√(1 + sinx) (1 – sinx)

Here a² – b² = (a + b) (a – b)

= (1 – sinx)/√(1 – sin²x)

Here 1 – sin²x = cos²x

= (1 – sinx)/√cos²x

= (1 – sinx)/cosx

= (1/cosx) – (sinx/cosx)

Here 1/cosx = secx, sinx/cosx = tanx

= secx – tanx