Question

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

Answer 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