Question

The value of 1-tan²30°/ 1+tan²30° is equal to ​

Answer 1

Trigonometry

The following are the tips and concept that can be use to find the solution:

• Having a basic knowledge of Trigonometric ratios and Angles.
• Trigonometric ratios are sin, cos, tan, cot, sec, cosec.
• The standard angles of these trigonometric ratios are 0°, 30°, 45°, 60° and 90°.

Analyse the values of important angles for all the six trigonometric ratios shown in the table given below:

$\Large{ \begin{tabular}{|c|c|c|c|c|c|} \cline{1-6} \theta & \sf 0^{\circ} & \sf 30^{\circ} & \sf 45^{\circ} & \sf 60^{\circ} & \sf 90^{\circ} \\ \cline{1-6} \sin & 0 & \dfrac{1}{2 } & \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} & 1 \\ \cline{1-6} \cos & 1 & \dfrac{ \sqrt{ 3 }}{2} } & \dfrac{1}{ \sqrt{2} } & \dfrac{ 1 }{ 2 } & 0 \\ \cline{1-6} \tan & 0 & \dfrac{1}{ \sqrt{3} } & 1 & \sqrt{3} & \infty \\ \cline{1-6} \cot & \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } &0 \\ \cline{1 - 6} \sec & 1 & \dfrac{2}{ \sqrt{3}} & \sqrt{2} & 2 & \infty \\ \cline{1-6} \csc & \infty & 2 & \sqrt{2 } & \dfrac{ 2 }{ \sqrt{ 3 } } & 1 \\ \cline{1 - 6}\end{tabular}}$

[Please check the attached image if the table is not visible]

Let's head to the question now.

$\implies \dfrac{1 - \tan^2 30^\circ}{1 + \tan^2 30^\circ} \\ \\ \implies \dfrac{1 - \left(\frac{1}{\sqrt{3}\right)^2}}{1 + \left(\frac{1}{\sqrt{3}\right)^2}} \\ \\ \implies \dfrac{1 - \frac{1}{3}}{1 + \frac{1}{3}} \\ \\ \implies \dfrac{\frac{3-1}{3}}{\frac{3+1}{3}} \\ \\ \implies \dfrac{1 - \frac{1}{3}}{1 + \frac{1}{3}} \\ \\ \implies \dfrac{\frac{2}{3}}{\frac{4}{3}} \\ \\ \implies \dfrac{\not2}{\not3} \times \dfrac{\not3}{\not4} \\ \\ \implies \dfrac{1}{2}$

Therefore,

$\implies \boxed{\dfrac{1 - \tan^2 30^\circ}{1 + \tan^2 30^\circ} = \drac{1}{2}}$

Hence, the required answer is 1/2.

Answer 2

Question

The value of 1-tan²30°/ 1+tan²30° is equal to

Solution,

$: \to \bold{ \frac{1 - tan ^{2} 30 \degree}{1 + {tan}^{2}30 \degree } } \\ \\ : \to \bold{1 - (\frac{1}{ \sqrt{3} }) ^{2} } \: \: \div \: \sf \: sec ^{2} \: 30 \degree \\ \\ : \to1 - \frac{1}{3} \: \: \div \: \: ( \frac{2}{ \sqrt{3} } ) ^{2} \\ \\ \: : \to \: \frac{3 - 1}{3} \: \: \div \: \: \frac{4}{3} \\ \\ \ : \to \cancel{ \frac{2}{3} } \: \times \: \cancel{ \frac{3}{4} } \\ \\ : \to \bold{ \red{ \frac{1}{2} }}$

Additional Information

What is Trigonometry?

= Trigonometry is branch of mathematics concerned with specific function of angle and their application to calculation there are six function of an angle commonly used in trigonometry their names and abbreviation.

1. Sine ( sin)

2. Cosine (cos)

3. Tangent (tan)

4. Cotangent (cot)

5. Secant (sec)

6. Cosecant (csc)