Question

Cot x +cosec x=5 what is the value of cos x?

Answer 1

given that
COTx+cosecx=5
(cosx/sinx)+cosecx=5
cosx=5sinx-1

Answer 2

The value of cos x is 12/13

Step-by-step explanation:

Consider the given expression cot x + cosec x = 5 as (1)

Since, the expression is value based, we also need to find the value for cos x.

Let us use the standard expressions to find the required value.

We know that,

$\csc ^{2} x-\cot ^{2} x=1$

We know that,

$(a+b)(a-b)=a^{2}-b^{2}$

$(\csc x-\cot x)(\csc x+\cot x)=1$

Substituting equation (1), we get,

$(\csc x-\cot x) \times 5=1$

$\csc x-\cot x=1 / 5 \rightarrow(2)$

Adding (1) and (2) gives

$\csc x+\cot x+\csc x-\cot x=5+\frac{1}{5}$

$2 \csc x=\frac{26}{5}$

$\csc x=\frac{13}{5}$

We know that,

$\csc x=\frac{1}{\sin x}$

Therefore,

$\sin x=\frac{5}{13}=\frac{\text { opposite side }}{\text { Hypotenuse }}$

Consider a right-angled triangle ABC, with AC = 13 and AB = 5  

By Pythagoras theorem,

$A C^{2}=A B^{2}+B C^{2}$

$13^{2}=5^{2}+B C^{2}$

$B C^{2}=169-25=144$

$BC = 12$

According to trigonometric identities, we get,

$\cos x=\frac{\text {Adjacent side}}{\text {Hypotenuse}}$

$\cos x=\frac{12}{13}$