Let $f$ be a real valued continuous function defined on $\mathbb{R}$ satisfying $$f'\left(\tan^{2}\theta\right) = \cos 2\theta + \tan\theta \sin 2\theta, \text{ for all real numbers } \theta.$$ If $f'(0) = -\cos\frac{\pi}{12}$ then find $f(1)$.
Let $f$ be a real valued continuous function defined on $\mathbb{R}$ satisfying
$$f'\left(\tan^{2}\theta\right) = \cos 2\theta + \tan\theta \sin 2\theta, \text{ for all real numbers } \theta.$$
If $f'(0) = -\cos\frac{\pi}{12}$ then find $f(1)$.