Question

In the following questions, two statements (i) and (ii) are given. Choose the valid statement.

(i) If sin θ + cosec θ = 4 then sin²θ + cosec²θ = 16

(ii) `sqrt((1 - cos^2θ)/(1 - sin^2θ)) = cot θ`

पर्याय

• Only (i)

• Only (ii)

• Both (i) and (ii)

• Neither (i) nor (ii)

Answer 1

Neither (i) nor (ii)

Explanation:

(i) Let x = sin θ.

Given `x + 1/x = 4`. 

Then `x^2 + 1/x^2 = (x + 1/x)^2 - 2`

= 4² – 2 

= 16 – 2

= 14

So, sin²θ + cosec²θ = 14, not 16.

Hence, (i) is false.

(ii) Simplify the left side:

`sqrt((1 - cos^2θ)/(1 - sin^2θ))`

= `sqrt(sin^2θ/cos^2θ)`

= |tan θ |, not cot θ which equals `1/(tan θ)`.

So, (ii) is not generally true.

Hence, (ii) is false.