If \(\tan^2\theta+\tan^4\theta=7\), then what is the value of \(7\cos^4\theta+\cos^2\theta\)?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 1
Taking \(\tan^2\theta\) common and using \(1+\tan^2\theta=\sec^2\theta\) writes the given value 7 as \(\tan^2\theta\sec^2\theta\). Putting this 7 into the expression makes the powers of \(\cos\theta\) cancel.
\(\tan^2\theta(1+\tan^2\theta)=\tan^2\theta\sec^2\theta=7\)
\(7\cos^4\theta=\tan^2\theta\sec^2\theta\cos^4\theta=\tan^2\theta\cos^2\theta=\sin^2\theta\)
\(7\cos^4\theta+\cos^2\theta=\sin^2\theta+\cos^2\theta=1\)
Such an angle exists: \(t=\tan^2\theta\) solves \(t^2+t-7=0\), whose positive root is \(t=\frac{-1+\sqrt{29}}{2}\approx 2.19\).