यदि tan θ = 2ab/(a² - b²), तो cos θ का मान क्या होगा?

Steps to Solve

1. Understanding the Given Information

We are given that ( \tan \theta = \frac{2ab}{a^2 - b^2} ). To find ( \cos \theta ), we need to use the relationship among the trigonometric functions.

2. Using the Identity for ( \tan \theta )

From the identity ( \cos^2 \theta + \sin^2 \theta = 1 ) and knowing that ( \tan \theta = \frac{\sin \theta}{\cos \theta} ), we can express ( \sin \theta ) in terms of ( \tan \theta ) and ( \cos \theta ):

$$ \sin \theta = \tan \theta \cdot \cos \theta $$

3. Finding ( \sin \theta )

Substituting the value of ( \tan \theta ),

$$ \sin \theta = \frac{2ab}{a^2 - b^2} \cdot \cos \theta $$

4. Substituting into the Pythagorean Identity

Plugging ( \sin \theta ) into the identity:

$$ \cos^2 \theta + \left(\frac{2ab \cdot \cos \theta}{a^2 - b^2}\right)^2 = 1 $$

5. Simplifying the Equation

Expanding the equation:

$$ \cos^2 \theta + \frac{4a^2b^2 \cos^2 \theta}{(a^2 - b^2)^2} = 1 $$

Factoring out ( \cos^2 \theta ):

$$ \cos^2 \theta \left( 1 + \frac{4a^2b^2}{(a^2 - b^2)^2} \right) = 1 $$

6. Final Expression for ( \cos^2 \theta )

Now solving for ( \cos^2 \theta ):

$$ \cos^2 \theta = \frac{1}{1 + \frac{4a^2b^2}{(a^2 - b^2)^2}} $$

7. Finding ( \cos \theta )

Taking the square root to find ( \cos \theta ):

$$ \cos \theta = \frac{(a^2 - b^2)}{\sqrt{(a^2 - b^2)^2 + 4a^2b^2}} $$