Question

simplify the trigonometric expression below: tanθ·sinθ·cosθ + cos2θ a) secθ holiday inn b) cscθ tigers stadium c) sin²θ home depot d) tan²θ farmers market e) cos²θ waffle house

Explanation:

Step1: Substitute $\tan\theta$

Since $\tan\theta=\frac{\sin\theta}{\cos\theta}$, the expression $\tan\theta\cdot\sin\theta\cdot\cos\theta+\cos2\theta$ becomes $\frac{\sin\theta}{\cos\theta}\cdot\sin\theta\cdot\cos\theta+\cos2\theta$.
Simplify $\frac{\sin\theta}{\cos\theta}\cdot\sin\theta\cdot\cos\theta$:
$$\frac{\sin\theta}{\cos\theta}\cdot\sin\theta\cdot\cos\theta=\sin^{2}\theta$$
So the expression is now $\sin^{2}\theta+\cos2\theta$.

Step2: Use double - angle formula

Recall the double - angle formula $\cos2\theta=\cos^{2}\theta - \sin^{2}\theta$.
Substitute it into $\sin^{2}\theta+\cos2\theta$:
$$\sin^{2}\theta+\cos^{2}\theta-\sin^{2}\theta$$

Step3: Simplify the expression

Combine like terms:
$$\sin^{2}\theta+\cos^{2}\theta-\sin^{2}\theta=\cos^{2}\theta$$

Answer:

E. $\cos^{2}\theta$