If cosec A - cot A = 4/5, then cosec A =a)47/40b)59/40c)51/40d)41/40Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Explanation :

Cosec A - cot A = 4 / 5 ----- (i)
Using identity :- cosec^2 - cot^2 A = 1
( a^2 - b^2 ) = ( a + b ) ( a - b )
(cosec A + cot A ) ( cosec A - cot A ) = 1
(cosec A + cot A ) 4 / 5 = 1
cosec A + cot A = 5 / 4 -----(ii)

Now ,

From (i) & (ii) equation, we get
cosecA - cotA = 4 / 5
cosec A + cot A = 5/ 4
2 cosec A = 4/ 5 + 5 / 4
2 cosec A = 16 + 25 / 20
2 cosec A = 41 / 20
cosec A = 41 / 20 x 2
cosec A = 41 / 40
Hence option (D) is correct answer .

You can read Important Definitions & Formulas of Trigonometry through the document:

Important Definitions & Formulas: Introduction to Trigonometry

https://edurev.in/studytube/important-definitions-formulas-introduction-to-tri/d87b442d-51bf-46ef-8d52-6f3a45b69468_t

Answer

CosecA-CotA=4/5---(i),
also cosec²A-cot²A=1,
so, (CosecA-CotA )(cosecA+CotA)=1,
4/5(CosecA+CotA)=1,
CosecA+CotA=4/5--(ii),
from (i) and (ii),
CosecA=41/40

Answer

Answer

Given, cosec A - cot A = 4/5
We need to find the value of cosec A.
Let's start by using the identity cosec A = 1/sin A and cot A = cos A/sin A.
Substituting these values in the given equation, we get:
1/sin A - cos A/sin A = 4/5
Multiplying both sides by sin A, we get:
1 - cos A = (4/5)sin A
Now, we can use the identity 1 - cos^2 A = sin^2 A to eliminate sin A from the equation.
Substituting this value, we get:
sin^2 A - cos^2 A = (4/5)sin A
Rearranging, we get:
sin^2 A - (4/5)sin A - cos^2 A = 0
Now, we can use the identity sin^2 A + cos^2 A = 1 to substitute cos^2 A in terms of sin^2 A.
Substituting this value, we get:
sin^2 A - (4/5)sin A - (1 - sin^2 A) = 0
Simplifying, we get:
2sin^2 A - (4/5)sin A - 1 = 0
Using the quadratic formula, we get:
sin A = (4/5 ± √(16/25 + 8))/4
sin A = (4/5 ± √(96/25))/4
sin A = (4/5 ± 4√6/5)/4
sin A = (1 ± √6)/5
Since cosec A = 1/sin A, we get:
cosec A = 5/(1 ± √6)
Since cosec A is positive, we take the positive value, and we get:
cosec A = 5/(1 + √6)
Simplifying this expression, we get:
cosec A = (5 - 5√6)/(-1 + 6)
cosec A = (5 - 5√6)/5
cosec A = 1 - √6
Therefore, the correct answer is option D, 41/40.

Answer

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