If (1/(1 - sin θ)) + (1/(1 + sin θ)) = 4 sec θ, where 0

Question

If (1/(1 - sin θ)) + (1/(1 + sin θ)) = 4 sec θ, where 0 < θ < 90°, find the value of (cot θ + csc θ).

Accepted Answer

Given equation: (1/(1 - sin θ)) + (1/(1 + sin θ)) = 4 sec θ

Step 1: Simplify the left-hand side (LHS):
LHS = (1/(1 - sin θ)) + (1/(1 + sin θ))
= [(1 + sin θ) + (1 - sin θ)] / [(1 - sin θ)(1 + sin θ)]
= (1 + sin θ + 1 - sin θ) / (1 - sin² θ)
= 2 / cos² θ

Step 2: The right-hand side (RHS) is 4 sec θ = 4 / cos θ

Step 3: Equate LHS and RHS:
2 / cos² θ = 4 / cos θ

Multiply both sides by cos² θ:
2 = 4 cos θ

Divide both sides by 4:
cos θ = 1/2

Step 4: Since 0 < θ < 90°, θ = 60°

Step 5: Find (cot θ + csc θ):

cot 60° = 1 / tan 60° = 1 / √3 = √3 / 3
csc 60° = 1 / sin 60° = 1 / (√3 / 2) = 2 / √3 = (2√3) / 3

Sum:
cot θ + csc θ = (√3 / 3) + (2√3 / 3) = (3√3) / 3 = √3

Therefore, the value is √3.

Answer

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