Question

if tanx=1/7,tany=1/3,prove that cos2x=sin4y​

Answer 1

Answer:

Given:

tan x = 1/7

tan y = 1/3

To Prove: cos 2x = sin 4y

We know that

tan x = opposite / adjacent = 1/7

So using these value in right angle triangle ABC and Pythagoras theorem,

AC² = 1² + 7² = 1 + 49 = 50

AC = √50 = 5√2

⇒ sin x = opposite/hypotenuse = 1/5√2

⇒ cos x = adjacent/hypotenuse = 7/5√2

Also,

tan y = opposite / adjacent = 1/3

So using these value in right angle triangle LMN and Pythagoras theorem,

AC² = 1² + 3² = 1 + 9 = 10

AC = √10

⇒ sin y = opposite/hypotenuse = 1/√10

⇒ cos y = adjacent/hypotenuse = 3/√10

Now,

LHS = cos 2x = 2cos² x -1 = 2(7/5√2)² - 1 = 2(49/50) - 1

       = 24/25

RHS = sin 4y = 2 sin 2y cos 2y

       = 4 sin y cos y (2cos² y -1)

       = 4 (1/√10)(3/√10) ( 2(3/√10)² - 1 )

       = (12/10) ( 9/5 - 1 )

       = (6/5) ( 4/5 )

       = 24/25

LHS = RHS

Hence proved.