CUET PG 2023 — Mathematics PYQ

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Question

CUET PG 2023 — Mathematics PYQ

CUET PG | Mathematics | 2023

If cot² 45°-sin² 45° = K sin² 30° x tan² 45° x sec² 45, then the value of K is

Choose the correct answer:

Explanation

1. Identify the Trigonometric Values

First, let's plug in the standard values for the trigonometric ratios:

$\cot 45^\circ = 1$
$\sin 45^\circ = \frac{1}{\sqrt{2}}$
$\sin 30^\circ = \frac{1}{2}$
$\tan 45^\circ = 1$
$\sec 45^\circ = \sqrt{2}$

2. Solve the Left Hand Side (LHS)

\text{LHS} = \cot^2 45^\circ - \sin^2 45^\circ

\text{LHS} = (1)^2 - \left(\frac{1}{\sqrt{2}}\right)^2

\text{LHS} = 1 - \frac{1}{2}

\text{LHS} = \frac{1}{2}

3. Solve the Right Hand Side (RHS)

\text{RHS} = K \cdot \sin^2 30^\circ \cdot \tan^2 45^\circ \cdot \sec^2 45^\circ

\text{RHS} = K \cdot \left(\frac{1}{2}\right)^2 \cdot (1)^2 \cdot (\sqrt{2})^2

\text{RHS} = K \cdot \frac{1}{4} \cdot 1 \cdot 2

\text{RHS} = K \cdot \frac{2}{4}

\text{RHS} = K \cdot \frac{1}{2}

4. Find the value of K

Equating LHS and RHS:

\frac{1}{2} = K \cdot \frac{1}{2}

Divide both sides by $\frac{1}{2}$ :

K = 1

The value of $K$ is $1$ .