Samikaran ko samjhen
Yadi \( \cos \theta = \frac{2}{3} \) diya gaya hai, to humein \( 2 \sec^2 \theta + 2 \tan^2 \theta - 9 \) ka maan gyat karna hai.

Trigonometric identities ka upyog
Trigonometric identities ka istemal karte hue, humein pata hai:
- \( \sec^2 \theta = 1 + \tan^2 \theta \)
Isse hum \( \sec^2 \theta \) ko \( \tan^2 \theta \) ke terms mein vyakt kar sakte hain.

Cosine aur sine ka sambandh
Pehle, \( \cos \theta = \frac{2}{3} \) se \( \sin^2 \theta \) ka maan gyat karte hain:
- \( \sin^2 \theta + \cos^2 \theta = 1 \)
- \( \sin^2 \theta = 1 - \cos^2 \theta \)
- \( \sin^2 \theta = 1 - \left(\frac{2}{3}\right)^2 = 1 - \frac{4}{9} = \frac{5}{9} \)
Ab, \( \sin \theta \) aur \( \tan \theta \) ka maan gyat karte hain:
- \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\sqrt{\frac{5}{9}}}{\frac{2}{3}} = \frac{\sqrt{5}}{2} \)

Tan aur sec ka maan gyat karna
- \( \tan^2 \theta = \left(\frac{\sqrt{5}}{2}\right)^2 = \frac{5}{4} \)
- \( \sec^2 \theta = 1 + \tan^2 \theta = 1 + \frac{5}{4} = \frac{9}{4} \)

Final samikaran
Ab, \( 2 \sec^2 \theta + 2 \tan^2 \theta - 9 \) ka maan gyat karte hain:
- \( 2 \sec^2 \theta = 2 \times \frac{9}{4} = \frac{18}{4} = \frac{9}{2} \)
- \( 2 \tan^2 \theta = 2 \times \frac{5}{4} = \frac{10}{4} = \frac{5}{2} \)
Is prakar:
\[
2 \sec^2 \theta + 2 \tan^2 \theta - 9 = \frac{9}{2} + \frac{5}{2} - 9 = \frac{14}{2} - 9 = 7 - 9 = -2
\]

Antim uttar
Is prakar, final maan hai \( -2 \).