The value of is equal to:

4 Explanation: 1. Group the Terms and Use Complementary Angles We can group the terms to use the identity . Rearranging the expression: Apply the complementary angle identity: Substitute these into the expression: 2. Use the Identity Recall that: Multiplying numerator and denominator by : 3. Apply the Identity to the Groups For the first group ( ): For the second group ( ): The expression now is: 4. Substitute Known Values Recall the standard values: Calculate the numerator: Calculate the denominator: 5. Final Calculation Correct Option: D)

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NIMCET 2021 — Mathematics PYQ

NIMCET | Mathematics | 2021

The value of $\tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ$ is equal to:

Choose the correct answer:

Explanation

1. Group the Terms and Use Complementary Angles

We can group the terms to use the identity $\tan(90^\circ - \theta) = \cot \theta$ .

Rearranging the expression:

$E = (\tan 81^\circ + \tan 9^\circ) - (\tan 63^\circ + \tan 27^\circ)$

Apply the complementary angle identity:

$\tan 81^\circ = \tan(90^\circ - 9^\circ) = \cot 9^\circ$

$\tan 63^\circ = \tan(90^\circ - 27^\circ) = \cot 27^\circ$

Substitute these into the expression:

$E = (\cot 9^\circ + \tan 9^\circ) - (\cot 27^\circ + \tan 27^\circ)$

2. Use the Identity $\tan \theta + \cot \theta = \frac{2}{\sin 2\theta}$

Recall that:

$\tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \frac{1}{\sin \theta \cos \theta}$

Multiplying numerator and denominator by $2$ :

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

3. Apply the Identity to the Groups

For the first group ( $\theta = 9^\circ$ ):

$\cot 9^\circ + \tan 9^\circ = \frac{2}{\sin 18^\circ}$

For the second group ( $\theta = 27^\circ$ ):

$\cot 27^\circ + \tan 27^\circ = \frac{2}{\sin 54^\circ}$

The expression now is:

$E = \frac{2}{\sin 18^\circ} - \frac{2}{\sin 54^\circ}$

$E = 2 \left( \frac{\sin 54^\circ - \sin 18^\circ}{\sin 54^\circ \sin 18^\circ} \right)$

4. Substitute Known Values

Recall the standard values:

$\sin 18^\circ = \frac{\sqrt{5} - 1}{4}$

$\sin 54^\circ = \cos 36^\circ = \frac{\sqrt{5} + 1}{4}$

Calculate the numerator:

$\sin 54^\circ - \sin 18^\circ = \frac{\sqrt{5} + 1}{4} - \frac{\sqrt{5} - 1}{4} = \frac{2}{4} = \frac{1}{2}$

Calculate the denominator:

$\sin 54^\circ \sin 18^\circ = \left(\frac{\sqrt{5} + 1}{4}\right) \left(\frac{\sqrt{5} - 1}{4}\right) = \frac{(\sqrt{5})^2 - 1^2}{16} = \frac{5 - 1}{16} = \frac{4}{16} = \frac{1}{4}$

5. Final Calculation

$E = 2 \left( \frac{1/2}{1/4} \right) = 2 \left( \frac{1}{2} \times 4 \right)$

$E = 2 \times 2 = 4$

Correct Option: D) $4$

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