If cosec A = 2 then obtain the value of cot A + (sin A /1+ cos A)

Solution:

Trigonometry helps us to understand the relation between the sides and angles of a right-angle triangle. Cosec, Sec and Cot are three of the six trigonometric ratios of a right-angled triangle. Let's look into the steps below.

Given: cosec A = 2

We know that,

cosec A=1 / sinA

⇒ sin A = 1/2.......................(i) 

Using trigonometric identity, we know that

sin²A + cos²A = 1..................(ii)

Now, put the value of sin A from equation (i) in equation (ii),

⇒ (1/2)² + cos²A = 1

⇒ 1/4 + cos²A = 1

⇒ 1 – 1/4 = cos²A

⇒ cos²A = 3/4

⇒ cos A = √3 / 2 ..................... (iii)

From trigonometric ratios we know that,

tan A = sin A / cos A ....................(iv)

Substituting the values of sin A from equation (i) and the value of cosA from equation (iii) in equation (iv) we get,

⇒ tan A = (1/2) × 2/√(3)

⇒ tan A = 1/√(3)

⇒ cot A = √(3) [ Since, cot A = 1/ tan A]................(v)

Now, let's evaluate

cot A + {sin A / (1+cosA)}

Substituting the values from (i), (iii) and (v) in (cot A + sinA) / (1+cosA) we get,

= √3 + {(1/2) / ( 1 + (√3/2))}

= √3 + {1 / (2 + √3)}

On evaluating,

= (2√3 + 4) / (2 + √3)

Rationalizing the denominator,

= (2√3 + 4) (2 - √3)  / (2 + √3) (2 - √3) 

= (4√3 + 8 - 4√3 - 6) / (4 - 3)

= 2

Thus, If cosec A = 2 then, the value of cot A + (sin A /1+ cos A) is 2

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If cosec A = 2 then obtain the value of cot A + (sin A /1+ cos A)

Summary:

If cosec A = 2 then, the value of cot A + (sin A /1+ cos A) is 2