If tan⁴ θ + tan² θ = 1, what is the value of 11(cos⁴ θ + cos² θ)?

Correct option is D

Given:

​$$\tan^4 \theta + \tan^2 \theta = 1$$​​

Formula Used:

​$$1 + \tan^2 \theta = \sec^2 \theta$$​​

Solution:

Start with the given equation:

​$$\tan^2 \theta (\tan^2 \theta + 1) = 1 \\\tan^2 \theta \times \sec^2 \theta = 1 \\\tan^2 \theta = \cos^2 \theta \\$$​​

Substitute and simplify:

​$$11(\cos^4 \theta + \cos^2 \theta) \\= 11 \cos^2 \theta (\cos^2 \theta + 1) \\$$​​

Substitute $$\cos^2 \theta = \tan^2 \theta :$$​​

​$$= 11 \cos^2 \theta (\tan^2 \theta + 1) \\= 11 \cos^2 \theta \sec^2 \theta \\= 11 \cos^2 \theta \times \frac{1}{\cos^2 \theta}= 11$$​​

The value is 11 .