Question

In problems 19-98, establish each identity.

$3{\sin }^2\left(\theta \right)+4{\cos }^2\left(\theta \right)=3+{\cos }^2\left(\theta \right)$

Short answer

We have proved the identity$3{\sin }^2\left(\theta \right)+4{\cos }^2\left(\theta \right)=3+{\cos }^2\left(\theta \right)$.

Step-by-step solution

Step 1. Given information.

We have:

$3{\sin }^2\left(\theta \right)+4{\cos }^2\left(\theta \right)=3+{\cos }^2\left(\theta \right)$

Step 2. Establish the identity.

Take the left hand side:

$3{\sin }^2\left(\theta \right)+4{\cos }^2\left(\theta \right)$

Apply identity: ${\sin }^2\left(\theta \right)=1-{\cos }^2\theta$

$$\begin{gathered} =3\left(1-{\cos }^2\left(\theta \right)\right)+4{\cos }^2\left(\theta \right) \\ =3-3{\cos }^2\theta +4{\cos }^2\left(\theta \right) \\ =3+{\cos }^2\left(\theta \right) \end{gathered}$$

Hence, proved.