Question

if tan(x+y)=a+b and tan(x-y)=a-b then prove that atan(x)-btan(y)= a^2-b^2​

Answer 1

Answer:

given tan[x+y] = a+b and tan[x-y] = a-b

⇒tanx+tany =a+b  and tanx-tany=a-b

Step-by-step explanation:

now add the both ,we get

tanx + tany =a+b

tanx  - tany =a-b

⇒tanx=a²

and do like this

Answer 2

★Given:-

• tan(x+y)=a+b
• tan(x-y)=a-b

★To prove:-

• atan(x)-btan(y)= a²-b²

★Proof:-

→tan(x+y)=a+b---(1)

→tan(x-y)=a-b----(2)

Multiplying equation (1)&(2),

⇒tan (x+y) tan (x-y) = (a+b)(a-b)

Using the formula,

★(a+b)(a-b)=a²-b²

⇒(tan²x-tan²y)/(1-tan²x tan²y)=a²-b²----(3)

⇒tan(x+y)+tan(x-y)=a+b+a-b

                             =2a------(4)

⇒(tanx+tany)/(1-tan xtan y) + (tanx-tany)/(1+tan xtan y)=2a

⇒2(tan x+tan xtan²y)/(1-tan²xtan²y)=2a

Multiplying both sides by tanx,

⇒atanx=(tan²x+tan²xtan²y)/(1-tan²xtan²y)-----(5)

⇒tan(x+y)-tan(x-y)=a+b-a+b

                             =2b

Now,

⇒btan y=(tan²y+tan²xtan²y)/(1-tan²xtan²y)----(4)

Using equation (3),

⇒atanx - btany=(tan²x-tan²y)/(1-tan²xtan²y)=a²-b²

Hence proved !

_________________