Question

The value of tan 60° – cot 60° is equal to the value of:

Options

• sec 30°

• cosec 30°

• cos 30°

• sin 60°

Answer 1

sec 30°

Explanation:

1. Evaluate the given expression

First, identify the trigonometric values for the angles in the expression:

`tan 60^circ = sqrt(3)`

`cot 60^circ = 1/sqrt(3)`

Now, subtract the two values:

`tan 60^circ -  cot 60^circ = sqrt(3) - 1/sqrt(3)`

To simplify, find a common denominator:

`sqrt(3) - 1/sqrt(3) = ((sqrt(3) xx sqrt(3)) - 1)/sqrt(3)`

= `(3 - 1)/sqrt(3)`

= `2/sqrt(3)`

2. Compare with options

Next, find the value of each provided option to see which matches `2/sqrt(3)`:

(A) sec 30°: Since `cos 30^circ = sqrt(3)/2`, then `sec 30^circ = 1/(cos 30^circ) = 2/sqrt(3)`.

(B) cosec 30°: Since `cos 30^circ = 1/2`, then `"cosec"  30^circ = 1/(sin 30^circ) = 2`.

(C) cos 30°: `cos 30^circ = sqrt(3)/2`.

(D) sin 30°: `sin 60^circ = sqrt(3)/2`.

The value of the original expression `(2/sqrt(3))` matches the value of sec 30°.