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How do you find the value of cot240 using double angle or half angle identity ?

Answer
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Hint: Here we are given with an angle with its trigonometric function. Actually 240 is a standard angle but we need to convert this angle in the basic angle with the help of double angle or half angle identity. Then we will use the angle with tangent function to find the value of cot240.

Complete step by step solution:
Given that cot240 is the angle whose value is to be found.
We know that tan2θ=2tanθ1tan2θ
So 240=2×120
So we can write
tan240=2tan1201tan2120
Also we can write 120=2×60
So the identity will be,
tan120=2tan601tan260
We know that tan60=3

Putting this value in the identity we get,
tan120=231(3)2
On calculating the square we get,
tan120=2313
tan120=232
On cancelling 2 we get,
tan120=3
Now we will put this value in the identity of 120
tan240=2(3)1(3)2

Taking the square,
tan240=2(3)13
tan240=2(3)2
Cancelling -2,
tan240=3
Now we know that
cot240=1tan240
So putting the value,
cot240=13

Hence, the value of cot240 is 13.

Note: We used half angle identity with double angle form. Also note that if we cannot reach the trigonometric function asked we need to take help of other such trigonometric functions that include or may help to reach the trigonometric function. We can go with the term like we know the value of nπ form for the same trigonometric function.