Question

if cosectheta+cottheta=k then prove that costheta=ksquare-1/ksquare+1​

Answer 1

Heya friend !

Given :-

cosec a + cot a = k --------------------------- (1)

Now,

We know, By identity,

$cosec^{2}a - cot^{2}a = 1$

or,

( cosec a - cot a ) ( cosec a + cot a)  = 1

( cosec a - cot a ) × k = 1

cosec a - cot a = $\frac{1}{k}$  ------------------------------ (2)

by (1) + (2), we get.

2 cosec a = $k + \frac{1}{k}$

2 cosec a = $\frac{k^{2} + 1 }{k}$

cosec a = $\frac{k^{2+1} }{2k}$ -------------------------------------(3)

Now, from equation (2),

cot a = $cosec a - \frac{1}{k}$

        = $\frac{k^{2} +1}{2k} - \frac{1}{k}\\\\\frac{k^{2}+1-2 }{2k}\\\\\frac{k^{2}-1 }{2k}$ ------------------------------------ (4)

Now, by (3) ÷ (4) , we get,

$\frac{cosec a}{cot a} = \frac{\frac{k^{2} + 1 }{2k} }{\frac{k^{2}-1 }{2k} }$

$\frac{\frac{1}{sina} }{\frac{cos a}{sina } }$ = $\frac{k^{2} +1}{k^{2}-1 }$

$\frac{1}{cosa}=\frac{k^{2}+1 }{k^{2}-1 }$

So,

cos a = $\frac{k^{2} -1}{k^{2}+1 }$

Hence Proved !

Thanks !

#BAL #answerwithquality