Question

If ; sec x + tan x = k obtain the values of cos x

Answer 1

sec x+tan x=k
1/cos x+ sin x/ cos x=k
Therefore cos x=( 1+sin x)/k

Answer 2

Answer:

The value of cos x is  $\cos x=\frac{2k}{k^2+1}$

Step-by-step explanation:

We have given that, $\sec x+\tan x=k$ ....(1)

To find : The values of $\cos x$ ?

Solution :

We know that,

$\sin^2x + \cos^2x = 1$

Dividing both side of expression with $\cos^2 x$,

$\tan^2x + 1 =\sec^2 x$

$\sec^2 x-\tan^2x=1$

Applying identity, $a^2-b^2=(a+b)(a-b)$

$(\sec x-\tan x)(\sec x+\tan x)=1$

$(\sec x-\tan x)(k)=1$  (Using (1))

$\sec x-\tan x=\frac{1}{k}$

From (1), $\tan x=k-\sec x$

$\sec x-(k-\sec x)=\frac{1}{k}$

$\sec x-k+\sec x=\frac{1}{k}$

$2\sec x=\frac{1}{k}+k$

$\sec x=\frac{1+k^2}{2k}$

We know, $\cos x=\frac{1}{\sec x}$

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

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

Therefore, The value of cos x is  $\cos x=\frac{2k}{k^2+1}$