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Class 10

>>Maths

>>Introduction to Trigonometry

>>Trigonometric Identities

>>If cos ec x - cot x = 32 , Find cos x

Question

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If cosecx−cotx=

2

3

, Find cosx In which quadrant does x lie ?

Medium

Solution

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Given,

cosecx−cotx=

2

3

……. (1)

We know that , cosec

2

x−cot

2

x=

2

3

(cosecx+cotx)(cosecx−cotx)=1 …..(2)

From equation (1) and(2),

(cosec x+cotx)

2

3

=1

cosec x+cotx=

3

2

Add equation (1) to (3) we get,

2cosecx=

3

2

+

2

3

cosec x=

12

13

sinx=

13

12

sinx is positive it means “sinx ” lie in either first or second quadrant

Putting cosec x=

12

13

in equation (3) we get

cosec x+cotx=

3

2

12

13

+cotx=

3

2

cot x=−

12

5

cotx is negative so can either be in second or fourth quadrant

but sinx is also positive x must lie in second quadrant

Now cotx=

sinx

cosx

cosx=cotx×sinx

cosx=−

12

5

×

13

12

cosx=−

12

5

Hence x lie in second quadrant.

Verified answer

Step-by-step explanation:

cosec x -cot x = 3/2

= 1-cos x / sin x = 3/ 2

= 2sin^2 (x/2) / (2 sin x /2 * cos x / 2)

= tan x / 2 = 3 /2

cos x = 1 - tan ^2 x /2 / (1 + tan ^2 x / 2)

cos x = 1 - 9 /4 / (1+ 9/4)

cos x = -5/13

it lies in the. second quadrant

Hope it helps!!!