Question

Value of tan 20° tan 40° tan 45°tan 50°tan70°​

Answer 1

Answer:

1

Step-by-step explanation:

$\tan \left(20^{\circ \:}\right)\tan \left(40^{\circ \:}\right)\tan \left(45^{\circ \:}\right)\tan \left(50^{\circ \:}\right)\tan \left(70^{\circ \:}\right)$

$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \tan \left(45^{\circ \:}\right)=1$

$=\tan \left(20^{\circ \:}\right)\tan \left(40^{\circ \:}\right)\cdot \:1\cdot \tan \left(50^{\circ \:}\right)\tan \left(70^{\circ \:}\right)$

$\mathrm{Simplify}$

$=\tan \left(20^{\circ \:}\right)\tan \left(40^{\circ \:}\right)\tan \left(50^{\circ \:}\right)\tan \left(70^{\circ \:}\right)$

$\tan \left(20^{\circ \:}\right)\tan \left(40^{\circ \:}\right)\tan \left(50^{\circ \:}\right)\tan \left(70^{\circ \:}\right)$

$\mathrm{Refine\:to\:a\:decimal\:form}$

$=1$

Alternative Method:

tan20° tan40° tan45° tan50° tan70°

​⇒  tan(90° - 20°) tan(90° - 40°) × 1 × tan50° tan70°

⇒ cot 70° cot 50° tan 50° tan 70°

Simplify;

1

Answer 2

Question:

Find the value of the given trigonometric problem:-

tan20°tan40°tan45°tan50°tan70°

Answer:

The value of the given trigonometric problem is 1.

Step-by-step explanation:

We know that 20°= 90°-70°

and also tan 40°=90°-50°

So the given equation can be written as

tan(90°-70°)tan(90°-50°)tan45°tan50°tan70°

Also we know that

tan(90°-theta) = cot theta.

Here let theta =70°

So tan (90°-70°)= cot 70°

Again let theta =50°

So tan(90°-50°)=cot 50°

So now we can write the given equation as:-

cot50°cot70°tan45°tan50°tan70°

Also the value of tan 45°= 1

We know that tan theta * cot theta =1

So the equation now becomes:-

1*1*1*1*1= 1