Question

Consider the following statements:1. If y=ln⁡(sec⁡x+tan⁡x), then dydx=sec⁡x.2. If y=ln⁡(cosecx−cot⁡x), then dydx=cosecx.Which of the above is / are correct?

• A. (A) 1 only
• B. (B) 2 only
• C. (C) Both 1 and 2
• D. (D) Neither 1 nor 2

Answer 1

Answer: (C)

(C) Both 1 and 2

Answer 2

Explanation:
Concept:y=ln⁡x then dydx=1xy=tan⁡x then dydx=sec2⁡xy=sec⁡x then dydx=sec⁡x⋅tan⁡xy=cosecx then dydx=−cosecx⋅cot⁡xy=cot⁡x then dydx=−cosec2xLet, y=ln⁡(sec⁡x+tan⁡x) ⇒dydx=1sec⁡x+tan⁡x×(sec⁡x⋅tan⁡x+sec2⁡x) ⇒dydx=sec⁡xsec⁡x+tan⁡x×(tan⁡x+sec⁡x) ⇒dydx=sec⁡x Statement (1) is true. Now, y=ln⁡(cosecx−cot⁡x),⇒dydx=1cosecx−cot⁡x×(−cosecx⋅cot⁡x+cosec2x) ⇒dydx=cosecxcosecx−cot⁡x×(cosecx.−cot⁡x) ⇒dydx=cosecx
Statement (2) is true.Hence, the correct option is (C).