Question

If $\sec x+\tan x=p$, then which of the following is/are correct?

• A. $\sin x=\frac{p^2-1}{p^2+1}$
• B. $\cos x=\frac{2p}{p^2+1}$
• C. $\tan x=\frac{p^2-1}{2p}$
• D. $\sec x=\frac{p^2+1}{2p}$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $\sin x=\frac{p^2-1}{p^2+1}$
B $\cos x=\frac{2p}{p^2+1}$
C $\tan x=\frac{p^2-1}{2p}$
D $\sec x=\frac{p^2+1}{2p}$

$\sec x+\tan x=p\cdots \left(1\right)$
We know that,
${\sec }^{2}x-{\tan }^{2}x=1\Rightarrow \sec x-\tan x=\frac{1}{\sec x+\tan x}\Rightarrow \sec x-\tan x=\frac{1}{p}\cdots \left(2\right)$
From equation $\left(1\right)$ and $\left(2\right)$,
$2\sec x=p+\frac{1}{p}\Rightarrow \sec x=\frac{p^2+1}{2p}\Rightarrow \cos x=\frac{2p}{p^2+1}$
And
$2\tan x=p-\frac{1}{p}\Rightarrow \tan x=\frac{p^2-1}{2p}$
Now,
$\sin x=\tan x\times \cos x=\frac{p^2-1}{p^2+1}$