If tan x=1/7 and tan y=1/3 …
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Gaurav Seth 5 years, 2 months ago
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.
1Thank You