JIPMAT 2025 (QA) - If + / - = 3, then value of ^4 - ^4 is | PYQs + Solutions

JIPMAT 2025

Geometry

Trigonometry

Hard

If $\frac{\sin\theta + \cos\theta}{\sin\theta - \cos\theta} = 3$ , then value of $\sin^4\theta - \cos^4\theta$ is

\frac{1}{5}

\frac{2}{5}

\frac{3}{5}

\frac{4}{5}

✅ Correct Option: 3

\frac{\sin\theta + \cos\theta}{\sin\theta - \cos\theta} = 3

{\sin\theta + \cos\theta} = 3({\sin\theta - \cos\theta})

2\cos\theta = 4{\sin\theta}

\cos\theta = 2 {\sin\theta}

Now,

\sin^4\theta - \cos^4\theta = (\sin^2\theta + \cos^2\theta) (\sin^2\theta - \cos^2\theta)

= (\sin^2\theta + \cos^2\theta) (\sin\theta + \cos\theta) (\sin\theta - \cos\theta)

Substituting:

\cos\theta = 2 {\sin\theta}

= (\sin^2\theta+ (2\sin\theta)^2) (\sin\theta + 2 sin\theta) (\sin\theta - 2 sin\theta)

\newline

= (5\sin^2\theta) (3\sin\theta ) (-\sin\theta )

\newline

= 15 \sin^4\theta

Now, we know

\sin^2\theta + \cos^2\theta = 1

\sin^2\theta + 4\sin^2\theta = 1

since,

\cos\theta = 2 {\sin\theta}

=5\sin^2\theta = 1

\newline

=\sin^2\theta = \frac{1}{5}

Thus,

15 \sin^4\theta = 15 \times (\frac{1}{5})^2 = \frac{15}{25} = \frac{3}{5}

Hence, the correct option is Option 3.