`I_(n)=int_(0)^(pi//4) tan^(n)x dx`, where `n in N`
Statement-1: `int_(0)^(pi//4) tan^(4)x dx=(3pi-8)/(12)`
Statement-2: `I_(n)+I_(n-2)=(1)/(n-1)`

Options

A. Statement-1 is true, Statement-2 is True,Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer

A

Explanation

We have,
`I_(n)-underset(0)overset(pi//4)int tan^(n)x dx`
`rArr I_(n-2)=underset(0)overset(pi//4)int tan^(n-2)x dx`
`I_(n)+I_(n-2)=underset(0)overset(pi//4)int(tan^(n)x+tan^(n-2)x)dx`
`rArr I_(n)+I_(n-2)=underset(0)overset(pi//4)int tan^(n-2)x sec^(2)x dx`
`rArr I_(n)+I_(n-2)=[(tan^(n-1))/(n-1)]_(0)^(pi//4)=(1)/(n-1)`
So, statement-2 is true.
Now,
`I_(n)+I_(n-2)=(1)/(n-1)`
`rArr I_(4)+I_(2)=(1)/(3)" "` [Replacing n by 4]
and, `I_(2)+I_(0)=1" "` [Replacing n by 2]
`rArr I_(4)+1-I_(0)=(1)/(3)rArr I_(4)+1-(pi)/(4)=(1)/(3)`
`rArr I_(4)=(pi)/(4)-(2)/(3)=(3pi-8)/(12)`
So, statement-1 is also true and statement-2 is a correct explanation for statement-1.