[tan(45+x) - tan(45-x)] / [tan(45+x) + tan(45-x)] = 2sinxcosx?

Question

[tan(45+x) - tan(45-x)] / [tan(45+x) + tan(45-x)] = 2sinxcosx

Answer 1

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Method - 1

......................

On LHS, writing tan as ( sin / cos ),

and taking ( cos. cos ) as LCM,

...Numerator

= [ sin(45+x). cos(45-x) - cos(45+x). sin(45-x) ] / [ cos(45+x). cos(45-x) ]

= sin [ (45+x) - (45-x) ] / [ ... ... ]

= sin 2x / [ ... ... ]

= 2 sin x. cos x / [ ... ... ] ................ (1)

Similarly,

... Denominator

= [ sin(45+x). cos(45-x) + cos(45+x). sin(45-x) ] / [ ... ... ]

= sin [ (45+x) + (45-x) ] / [ ... ... ]

= sin 90 / [ ... ... ]

= 1 / [ ... ... ] ........................ (2)

Dividing (1) by (2),

LHS = 2 sin x. cos x = RHS ............ Q.E.D.

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Method - 2

....................................

Let t = tan x. Then,

tan (45°+x) = (1+t) / (1-t) and tan (45°-x) = (1-t) / (1+t).

Hence, on the LHS

... Numerator

= [ (1+t) / (1-t) ] - [ (1-t) / (1+t) ]

= [ (1+t)² - (1-t)² ] / (1-t²)

= 4t / (1-t²) .................................... (1)

Similarly, we can show that

Den. = 2(1+t²) / (1-t²) ......................(2)

Hence, from (1) and (2),

LHS = 4t / 2(1+t²) = 2. tan x / (1+tan² x) = 2.tan x / ( sec² x )

. . . .= 2( sin x / cos x ). ( cos² x ) = 2. sin x. cos x

. . . .= RHS

Hence, the result.

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Answer 2

i tried, i really did and im supposed to be good at this.. so r u sure the question is right?? i know u definitely multiply the brackets lol.. p.s i don't understand the other guy's answer..

Answer 3

Tangent=sine/cosine, tan(90+x)=1/tan(x) and sine(x)=cosine(45-x) and cosine(x)=sine(45-x). From this we get

(cos(45-x)/sin(45-x)-sin(45-x)/cos(45-x))/(cos(45-x)/sin(45-x)+sin(45-x)/cos(45-x))

then

(sin(x)/cos(x)-sin(x)/cos(x))/(sin(x)/cos(x)+cos(x)/sin(x))

=2*sin(x)*cos(x)