CUET PG 2022 — Mathematics PYQ

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Question

CUET PG 2022 — Mathematics PYQ

CUET PG | Mathematics | 2022

What is the value of $[tan^{2} (90 - θ) - sin^{2} (90 - θ)] cose c^{2} (90 - θ) cot^{2} (90 - θ)$

Choose the correct answer:

Explanation

1. Apply Complementary Angle Identities:

We know that:

$\tan(90-\theta) = \cot\theta$
$\sin(90-\theta) = \cos\theta$
$\csc(90-\theta) = \sec\theta$
$\cot(90-\theta) = \tan\theta$

Substituting these values into the expression:

[\cot^2\theta - \cos^2\theta] \cdot \sec^2\theta \cdot \tan^2\theta

2. Simplify the term inside the bracket:

We can write $\cot^2\theta$ as $\frac{\cos^2\theta}{\sin^2\theta}$ :

\cot^2\theta - \cos^2\theta = \frac{\cos^2\theta}{\sin^2\theta} - \cos^2\theta

Factor out $\cos^2\theta$ :

= \cos^2\theta \left( \frac{1}{\sin^2\theta} - 1 \right)

= \cos^2\theta (\csc^2\theta - 1)

Using the identity $\csc^2\theta - 1 = \cot^2\theta$ :

= \cos^2\theta \cdot \cot^2\theta

3. Substitute the simplified bracket back into the expression:

Now the expression becomes:

(\cos^2\theta \cdot \cot^2\theta) \cdot \sec^2\theta \cdot \tan^2\theta

4. Group the reciprocal terms:

Rearranging the terms:

(\cos^2\theta \cdot \sec^2\theta) \cdot (\cot^2\theta \cdot \tan^2\theta)

Since $\cos\theta \cdot \sec\theta = 1$ and $\cot\theta \cdot \tan\theta = 1$ :

(1)^2 \cdot (1)^2

= 1 \times 1 = 1