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

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

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  • Hemant
    Lv 7
    1 decade ago
    Favorite Answer

    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.

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

    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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    Happy To Help !

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  • 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..

  • 1 decade ago

    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)

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