Question

The value of tan 1° tan 2° tan 3° ………. tan 89° is

• A. 0
• B. 1
• C. \(\frac{1}{2}\)
• D. -1

Answer 1

The Correct Option is B

The given problem asks for the value of:
\(\tan 1^\circ \times \tan 2^\circ \times \tan 3^\circ \times \dots \times \tan 89^\circ\)
We can use the key trigonometric identity:
\[ \tan(90^\circ - x) = \cot x \]
So, for each pair \( \tan x^\circ \) and \( \tan(90^\circ - x^\circ) = \cot x^\circ \), their product is:
\[ \tan x^\circ \times \tan(90^\circ - x^\circ) = \tan x^\circ \times \cot x^\circ = 1 \]
Therefore, the product of tangents from \( 1^\circ \) to \( 89^\circ \) can be paired as:
\[ (\tan 1^\circ \times \tan 89^\circ), (\tan 2^\circ \times \tan 88^\circ), \dots \]
Each pair gives a product of 1, and since the number of terms is odd (with \( \tan 45^\circ \) in the middle), the value of the entire product is: \[ 1 \]The correct answer is (B) : 1.

Answer 2

We need to find the value of the product:

\[ P = \tan(1^\circ) \tan(2^\circ) \tan(3^\circ) \dots \tan(87^\circ) \tan(88^\circ) \tan(89^\circ) \]

We can use the trigonometric identity for complementary angles:

\[ \tan(90^\circ - \theta) = \cot(\theta) \]

Also, we know that \( \cot(\theta) = \frac{1}{\tan(\theta)} \). Therefore,

\[ \tan(\theta) \cot(\theta) = \tan(\theta) \times \frac{1}{\tan(\theta)} = 1 \]

Combining these, we get: 

\[ \tan(\theta) \tan(90^\circ - \theta) = \tan(\theta) \cot(\theta) = 1 \]

Now, let's pair the terms in the product \( P \) from the beginning and the end:

• \( \tan(1^\circ) \) pairs with \( \tan(89^\circ) = \tan(90^\circ - 1^\circ) = \cot(1^\circ) \)
• \( \tan(2^\circ) \) pairs with \( \tan(88^\circ) = \tan(90^\circ - 2^\circ) = \cot(2^\circ) \)
• \( \tan(3^\circ) \) pairs with \( \tan(87^\circ) = \tan(90^\circ - 3^\circ) = \cot(3^\circ) \)
• ...
• \( \tan(44^\circ) \) pairs with \( \tan(46^\circ) = \tan(90^\circ - 44^\circ) = \cot(44^\circ) \)

The product can be rewritten by grouping these pairs:

\[ P = [\tan(1^\circ) \tan(89^\circ)] \times [\tan(2^\circ) \tan(88^\circ)] \times \dots \times [\tan(44^\circ) \tan(46^\circ)] \times \tan(45^\circ) \]

Using the identity \( \tan(\theta) \tan(90^\circ - \theta) = 1 \), each pair in the brackets evaluates to 1:

\[ [\tan(1^\circ) \cot(1^\circ)] \times [\tan(2^\circ) \cot(2^\circ)] \times \dots \times [\tan(44^\circ) \cot(44^\circ)] \times \tan(45^\circ) \]

\[ P = (1) \times (1) \times \dots \times (1) \times \tan(45^\circ) \]

There are 44 such pairs, all equal to 1.

The middle term is \( \tan(45^\circ) \).

We know that \( \tan(45^\circ) = 1 \).

So, the product becomes:

\[ P = 1 \times 1 \times \dots \times 1 \times 1 \]

\[ P = 1 \]

Therefore, the value of the expression is 1.

Comparing this result with the given options:

• 0
• 1
• \( \frac{1}{2} \)
• -1

The correct option is 1.