Question

14. The value of (tan 1°. tan 2º . tan 3° ... tan 899) = (a) 1 (6) O (c) 1 (d) 3 oo Y​

Answer 1

Answer:

1--> a)

Step-by-step explanation:

tan 1° tan 2° tan 3° … tan 89° = [tan 1° tan 2° … tan 44°] tan 45° [tan (90° – 44°) tan (90° – 43°) … tan (90° – 1°)] = [tan 1° tan 2° … tan 44°] [cot 44° cot 43°……. cot 1°] = [tan 1°*cot 1°* tan 2°*cot 2°.... ]  (tan 1°*cot 1° = 1)

Answer 2

Trigonometric function

Let me write the question once for a better clarity.

Appropriate question:

The value of (tan(1°) tan(2°) tan(3°) ... tan(89°)) is

(a) 0

(b) 1

(c) ∞

(d) none

Step-by-step explanation:

The given expression is,

$\longrightarrow \tan(1^\circ) \tan(2^\circ) \tan(3^\circ) ... \tan(89^\circ)$

We need to find the value of above expression.

Let's solve the expression and understand the steps to get our final result.

$\implies \tan(1^\circ) \tan(2^\circ) ... \tan(45^\circ) ... \tan(88^\circ) \tan(89^\circ)$

$\implies \tan(1^\circ) \tan(2^\circ) ... \tan(45^\circ) ... \tan(90^\circ - 2^\circ) \tan(90^\circ - 1^\circ)$

We know that, $\tan(90) - x = \cot(x)$. So using this identity in the above equation, we obtain:

$\implies \tan(1^\circ) \tan(2^\circ) ... \tan(45^\circ) \cot(2^\circ) \cot(1^\circ)$

$\implies \tan(1^\circ) \cot(1^\circ) \tan(2^\circ) \cot(2^\circ) ... \tan(45^\circ)$

We know that, $\tan(x) \cot(x) = 1$. So using this identity in the above equation, we obtain the following results:

$\implies 1 \times 1 \times 1 \times \times \times ... \tan(45^\circ)$

We know that, $\tan(45^\circ) = 1$. So using this trigonometric angle formula in the above equation, we get the following results:

$\implies 1 \times 1 \times 1 \times 1 \times 1 \times ... 1$

$\implies 1$

Hence we arrived at a result:

$\boxed{\tan(1^\circ) \tan(2^\circ) \tan(3^\circ) ... \tan(89^\circ) = 1}$

$\rule{300}{2}$

Formulae used:

The following are the identities/formulas that have been used to find the solution:

$\boxed{\begin{array}{l}\bullet \; \; \tan(90) - x = \cot(x) \\ \bullet \; \; \tan(x) \cot(x) = 1 \\ \bullet \; \;\tan(45^\circ) = 1\end{array}}$