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Math Help - prove it

  1. #1
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    prove it

    sin theta(1 + tan theta) + cos theta(1 + cot theta) = sec theta + cosec theta

    plzz complete it as i am new to this chapter
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  2. #2
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    Quote Originally Posted by saha.subham View Post
    sin theta(1 + tan theta) + cos theta(1 + cot theta) = sec theta + cosec theta

    plzz complete it as i am new to this chapter

    It is just for easier input , i put  \theta = x so what we are going to prove is :

     \sin{x} ( 1 + \tan{x} ) + \cos{x} ( 1+ \cot{x} ) = \sec{x} + \csc{x}

    Consider

     \tan{x} = \frac{\sin{x}}{\cos{x}} and

     \cot{x} = \frac{\cos{x}}{\sin{x}}

    So

     1 + \tan{x} = 1 + \frac{\sin{x}}{\cos{x}}  = \frac{ \sin{x} + \cos{x} }{ \cos{x}}

     1 +  \cot{x} = 1 +  \frac{\cos{x}}{\sin{x}} = \frac{ \sin{x} + \cos{x} }{ \sin{x}}

    and we have


     \sin{x} ( 1 + \tan{x} ) + \cos{x} ( 1+ \cot{x} ) =

      \sin{x}~\frac{ \sin{x} + \cos{x} }{ \cos{x}} + \cos{x} ~ \frac{ \sin{x} + \cos{x} }{ \sin{x}}


    Look at the common factor :  \sin{x} + \cos{x} get them out and we obtain :

     (\sin{x} + \cos{x}) \left(\frac{ \sin{x}}{\cos{x}} + \frac{ \cos{x}}{\sin{x}} \right)


     = (\sin{x} + \cos{x}) ~ \frac{ \sin^2{x} + \cos^2{x} }{\sin{x}\cos{x}}

    I think you can finish it starting from here .
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  3. #3
    Senior Member nikhil's Avatar
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    Lightbulb another way

    sinx(1+tanx)+cosx(1+cotx)
    =sinx(1+tanx)+cosx[(1+tanx)/tanx]............................ [cotx=1/tanx]
    =(1+tanx)[sinx+(cosx/tanx)]
    =(1+tanx)[(sinx)^2+(cosx)^2]/sinx............................. [tanx=sinx/cosx]
    =(1+tanx)/sinx
    =((1/sinx)+(tanx/sinx))
    =secx+cosecx
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  4. #4
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