If cosecA=2 then obtain the value of cotA+sinA/1+cosA

Given

CosecA = 2

We need to find the value of cotA+sinA/1+cosA

Solution

Sin A = 1/2————(i) {sin A= 1/Cosec A}

By trigonometric identity we know that

cos²A + sin²A = 1———————(ii)

Substituting thevalue of Sin A from equation (i) in equation (ii)
cos²A + (1/2)² = 1

cos²A + 1/4 = 1
1 – 1/4 = cos²A

cos²A = 3/4

cosA = √(3)/2———–(iii)

We know that
TanA = sinA/CosA ————–(iv)

Substituting (i) and (iii) in (iv) we get,

So Tan A = 1/2 × 2/√(3)

TanA = 1/√(3)

CotA = √(3) (cot A = 1/ Tan A}—————-(v)

To find

cotA+sinA/1+cosA

substitute (i) (iii) and (v) in the above equation we get,

= √(3) + 1/2 / 1 + √(3)/2

= 2√(3) + 1/2 / 2 + √(3)/ 2

= 2√(3) + 1/2 × 2/2 + √(3)

= [2√(3) + 1] × [2 + √(3)]

= 2[2√(3) + 1] + √(3)[2√(3) + 1]

= 4√(3) + 2 + (2×3) + √(3)

= 5√(3) + 2 + 6

= 5√(3) + 8

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