Question

Consider the following statements:
(1) If $y=\ln (\sec x+\tan x)$, then $\frac{dy}{dx}=\sec x$
(2) If $y=\ln (\text{cosec}x-\cot x)$, then $\frac{dy}{dx}=\text{cosec}x$,
which of the above is/are correct?

• A. (1) only
• B. (2) only
• C. Both (1) and (2)
• D. Neither (1) and (2)

Answer 1

The correct option is C Both (1) and (2)

$y=\ln (\sec x+\tan x)$

Differentiating both the sides, we get

$\frac{dy}{dx}=\frac{1}{\sec x+\tan x}\cdot (\tan x\sec x+{\sec }^{2}x)$

$=\sec x$

$y=\ln (\text{cosec}x-\cot x)$

Differentiating both the sides, we get

$\frac{dy}{dx}=\frac{1}{\text{cosec}x-\cot x}\cdot (-\cot x\text{cosec}x+{\text{cosec}}^{2}x)=\text{cosec}x$

Thus, both statements are correct.
Hence, C is correct.