Question

If tan (x+y ).tan (x-y)=1 Then tan (2x/3)is

Answer 1

Answer:

1/√3

Step-by-step explanation:

tan(x+y).tan(x-y)=1

tan(x+y)=1/tan(x-y)

tan(x+y)=cot(x-y)

tan(x+y)=tan(90-(x-y))

x+y=90-x+y

2x+90

x=45

tan(2x/3)= tan((2×45)/3)

tan30= 1/√3

Answer 2

$\bold {tan(\frac{2x}{3}) = \frac{1}{\sqrt 3}}$

Step-by-step explanation:

Given,

tan(x+y).tan(x-y)=1 ........(1)

We have to find the value of $tan(\frac{2x}{3})$

From equation (1),

$tan(x+y) = \frac{1}{tan(x-y)}$

⇒tan(x+y) = cot(x-y)

⇒$tan(x+y) = tan(90-(x-y))\ [tan(90-\theta) = tan\theta]$

On comparison,

x+y=90-(x-y)

⇒x+y=90-x+y

⇒x = 90-x

⇒x+x=90

⇒2x=90

⇒$x = \frac{90}{2}$

            = 45°

So,$tan(\frac{2x}{3})$

= $tan(\frac{2\times 45}{3})$

= $tan (\frac{90}{3})$

= tan 30°

=$\frac{1}{\sqrt 3}$   [value of tan30°]

$\bold {Hence,tan(\frac{2x}{3}) = \frac{1}{\sqrt 3}}$