\((\tan\theta+2)(2\tan\theta+1)=5\tan\theta+\sec^2\theta\).
False.
Step 1: Expand the left-hand side (LHS).
LHS = \((\tan\theta + 2)(2\tan\theta + 1)\)
Use distributive property: \(a(b+c) = ab + ac\).
= \(\tan\theta(2\tan\theta + 1) + 2(2\tan\theta + 1)\)
= \(2\tan^2\theta + \tan\theta + 4\tan\theta + 2\)
= \(2\tan^2\theta + 5\tan\theta + 2\).
Step 2: Compare with the right-hand side (RHS).
RHS = \(5\tan\theta + \sec^2\theta\).
Step 3: Simplify the LHS using identity.
We know the trigonometric identity: \(1 + \tan^2\theta = \sec^2\theta\).
So, \(2\tan^2\theta + 2 = 2(\tan^2\theta + 1) = 2\sec^2\theta\).
Therefore, LHS = \(5\tan\theta + 2\sec^2\theta\).
Step 4: Final comparison.
LHS = \(5\tan\theta + 2\sec^2\theta\)
RHS = \(5\tan\theta + \sec^2\theta\)
Since \(2\sec^2\theta \neq \sec^2\theta\), the two sides are not equal.
Therefore, the given statement is False.