TG EAMCET 3 May 2025 Shift 2 Paper

The equation ax2+bx+c=0 has the roots tan‌θ and cot‌θ.
For a quadratic equation ax2+bx+c=0, if p and q are roots, then:
Sum of roots: p+q=−‌

b

a

Product of roots: pq=‌

c

a

Here, the roots are tan‌θ and cot‌θ. So, tan‌θ+cot‌θ=−‌

b

a

and tan‌θ⋅cot‌θ=‌

c

a

.
tan‌θ+cot‌θ=‌

sin‌θ

cos‌θ

+‌

cos‌θ

sin‌θ

This simplifies to: ‌

sin‌2θ+cos2θ

sin‌θ‌cos‌θ

=‌

1

sin‌θ‌cos‌θ

, because sin‌2θ+cos2θ=1
Now, tan‌θ+cot‌θ=‌

1

sin‌θ‌cos‌θ

But from earlier, tan‌θ+cot‌θ=−‌

b

a

.
So, ‌

1

sin‌θ‌cos‌θ

=−‌

b

a

.
For tan‌θ⋅cot‌θ=‌

c

a

:
tan‌θ⋅cot‌θ=1,‌ so ‌‌

c

a

=1⟹c=a.
Now, take the result ‌

1

sin‌θ‌cos‌θ

=−‌

b

a

.
Flip both sides: sin‌θ‌cos‌θ=−‌

a

b

2sin‌θ‌cos‌θ=2×(−‌

a

b

)=−‌

2a

b

But 2sin‌θ‌cos‌θ=sin‌2θ.
So, sin‌2θ=−‌

2a

b

.
Since a=c, we can also write sin‌2θ=−‌

2c

b

.