Prove that:
cos theta/1- tan theta+ Sin theta/ 1-cot theta
​

1 thought on “Prove that:<br />cos theta/1- tan theta+ Sin theta/ 1-cot theta<br />​”

1. Answer:

   Let I

   n

   =∫

   (secx+tanx)

   n

   sec

   2

   x

   dx (for n>1)

   ∴I

   n

   =∫

   (

   1+tan

   2

   x

   +tanx)

   n

   sec

   2

   x

   dx (as sec

   2

   θ=1+tan

   2

   θ)

   Now, let t=tanx⇒dt=sec

   2

   xdx, we then have

   I

   n

   =∫

   (

   1+t

   2

   +t)

   n

   1

   dt

   Making a hyperbolic substitution t=sinhy⟹dt=coshydy , we have

   I

   n

   =∫

   (

   1+sinh

   2

   y

   +sinhy)

   n

   coshy

   dy

   Now, as sinhy=

   2

   e

   y

   −e

   −y

   and 1+sinh

   2

   y=cosh

   2

   y, we get

   I

   n

   =

   2

   1

   ∫

   e

   ny

   e

   y

   +e

   −y

   dy

   ∴I

   n

   =

   2

   1

   ∫[e

   −(n−1)y

   +e

   −(n+1)y

   ]dy

   ∴I

   n

   =−

   2(n−1)

   e

   −(n−1)y

   −

   2(n+1)

   e

   −(n+1)y

   +C

   Now, re-substitute coshy−sinhy=e

   −y

   and sinhy=tanx, coshy=secx, we have

   I

   n

   =−

   2(n−1)

   (secx−tanx)

   (n−1)

   −

   2(n+1)

   (secx−tanx)

   (n+1)

   +C

   Applying the limits, we get

   I

   n

   =[−

   2(n−1)

   (secx−tanx)

   (n−1)

   −

   2(n+1)

   (secx−tanx)

   (n+1)

   ]

   0

   π/2

   ∴I

   n

   =

   n

   2

   −1

   n

   (Note that lim

   x→∞

   secx−tanx=0)

   Hence, proved

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