Question

Prove that cosA/(1-sinA)+cosA/(1+sinA) = 2secA

Answer 1

LHS
[cosA/(1-sinA)] + [cosA/(1+sinA)

Taking LCM
[cosA(1+sinA) + cosA(1-sinA)]/[(1-sinA)(1+sinA)]
=(cosA+cotA + cosA - cotA)/(1-sin²A)
=2cosA/cos²A
=2/cosA
=2secA
RHS

Answer 2

Concept

Trigonometry has three fundamental operations: sine, cosine, and tangent. The cotangent, secant, and cosecant are three crucial trigonometric functions that can be derived from these three fundamental ratios or functions.

Given

cosA/(1₋sinA) ₊ cosA/(1 ₊ sinA) = 2 secA

Find

we are asked to prove the expression cosA/(1₋sinA) ₊ cosA/(1 ₊ sinA) = 2 secA

Solution

given,  cos A/(1₋sin A) ₊ cos A/(1 ₊ sin A) = 2 sec A

L.H.S =  cos A/(1₋sin A) ₊ cos A/(1 ₊ sin A)

= cos A ₋ cos A sin A ₊ cos A ₊ cos A sin A / (1 ₋ sin A)(1 ₊ sin A)

= 2 cos A/ 1 ₋ sin²A  [∵ (a ₊ b)(a ₋ b) = a² ₋ b²]

= 2 cos A / cos²A  [ ∵ cos²A ₊ sin²A = 1 ⇒ cos²A = 1 ₋ sin²A]

= 2 / cos A [ 1/cos A = sec A]

= 2 sec A = R.H.S

Hence, proved.

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