Question

Evaluate:3 cot 31°.tan15°.cot 27°.tan75°.cot 63°.cot 59°.

Answer 1

By using above method, it can be solved by this way.

Answer 2

Given:

The trigonometric expression $3 \ \cot 31 \ \tan 15 \ \cot 27 \ \tan75 \ \cot 63 \ \cot59$.

To Find:

The value of the given expression.

Solution:

Since, we are familiar with the trigonometric properties:

$\tan x \cot x = 1\\\\\tan(90-x) = \cot x\\\\\& \ \cot(90-x) = \tan x$

We are going to use them to solve the given trigonometric expression.

$3 \ \cot 31 \ \tan 15 \ \cot 27 \ \tan75 \ \cot 63 \ \cot59$

After shifting so as to keep the trigonometric ratios with sum of angles as 90 close to each other, we get:

$= 3 \ \cot 31 \ \cot59 \ \tan 15 \ \tan75 \ \cot 63 \ \cot 27 \\\\ = 3 \ \cot (90-59) \ \cot59 \ \tan (90-75) \ \tan75 \ \cot (90-27) \ \cot 27$

Now, we will use the trigonometric properties mentioned above,

$= 3 \ \tan59 \ \cot59 \ \cot75 \ \tan75 \ \tan27 \ \cot 27$

Also, using tanx cotx = 1

$= 3 \times 1 \times 1 \times 1 = 3$

Thus, the value of the given trigonometric expression $3 \ \cot 31 \ \tan 15 \ \cot 27 \ \tan75 \ \cot 63 \ \cot59 = 1$.