UPSC NDA I 2025 Mathematics Paper

We are given:
‌p=tan(2α)−tan(α)
‌q=cot(α)−cot(2α)
We need to find tan2(α) in terms of p and q.
We start by simplifying the expression ‌

p

p+2q

‌

p

p+2q

=‌

1

1+‌ 2q p

This simplifies further as:
‌

1

1+‌ 2 tan(α)⋅tan(2α)

Now, simplify the fraction inside the denominator:
=‌

1

1+‌ 2 tan(α)⋅tan(2α)

=‌

1

1+‌ 1 2 ⋅tan(α)⋅tan(2α)

Now, expand both terms using trigonometric identities:
=‌

sin‌(α)⋅sin‌(2α)

sin‌(α)⋅cos(α)+cos(α)⋅cos(2α)

Simplifying this expression gives: =tan2(α)