Recall that cot is the reciprocal of tan

So, if cot 2x = 0 , then 1 / tan 2x = 0

Now for this fraction, the numerator is 1 and and denominator is tan 2x


When the denominator tan 2x gets bigger, 1 / tan 2x gets smaller and smaller.

So as tan 2x approaches infinity , 1 / tan 2x approaches 0 .

This is denoted by the following expressions :

tan 2x → ∞

1 / tan 2x → 0
cot 2x → 0

So if we have 1 / tan 2x = 0, then we can infer that tan 2x 's value is taken to be infinity.


And this occurs when 2x = π/2 (or in degreees, 2x = 90°)


Recall that for tan π/2 (or tan 90°) ,we basically have no right angled triangle at all since the adjacent side has no length

And since tan is opposite ÷ adjacent , we can say that when the angle = π/2 , tan π/2 is infinity since the denominator is 0

(Although strictly speaking, division by 0 is undefined and not infinity. But for the purposes of trigo, we take it to be so)


So when we are taking the inverse cotangent of 0 find 2x,

cot-¹ 0

We are actually doing the equivalent of tan-¹ (∞) = π/2 rad

(note that writing it this way is not really correct)

Alternatively, cot 2x = cos 2x / sin 2x

When cot 2x = 0, cos 2x / sin 2x = 0

The denominator cannot be 0 since division by 0 is undefined. So,

cos 2x = 0 × sin 2x

This means the numerator cos 2x = 0

And cos 2x = 0 when 2x = π/2 rad (or 90°)

(That's why we will write 2x = cos-¹ (0) = π/2)



Now,

Why do we see a kπ?


Because, the periodic nature of the cosine function means that for every addition to or subtraction of an integer multiple of π radians (or 180°) from π/2, the value of cos 2x will remain the same at 0.

It repeats every π radians for cos 2x = 0.

(For other values of cos 2x, you'll need to look at graph to determine the periodicity.)


So this is denoted by :

2x = π/2 + kπ, where k is an integer.


Example :


When k = 1, cos 2x = cos (π/2 + π) = cos (3π/2) = cos (π/2) = 0

When k = 2 cos 2x = cos (π/2 + 2π) = cos (5π/2) = cos (π/2) = 0


Basically ,

cos (-5π/2) = cos (-3π/2) = cos (-π/2) = cos (π/2) = cos (3π/2) = cos (5π/2) = cos (7π/2) = ...

This is neatly condensed into the equation 2x = π/2 + kπ, k is an integer.


Likewise, if we did it the cot-¹(0) way, the periodicity is also π radians.

Lastly, why do we see a 0 in replacement of kπ?



x = ½(π/2 + kπ)

x = π/4 + kπ/2


In the answer key, the provider just set k = 0 to demonstrate that he/she is just finding the basic angle of π/4 radians (or 45°)



It would be better to leave it as :


x = π/4 + kπ/2 , k is an integer

Or

x = π/4 (1 + 2k), k is an integer