Question

simplify the expression \\(\frac{\sec^2 \theta}{\cot^2 \theta + 1}\\) select one: \\(\circ\\) a. \\(\cos^2 \theta\\) \\(\circ\\) b. \\(\tan^2 \theta\\) \\(\circ\\) c. \\(\sin^2 \theta\\) \\(\circ\\) d. -1

Explanation:

Step1: Use Pythagorean identity for cotangent

Recall that $\cot^{2}\theta + 1=\csc^{2}\theta$. So the denominator becomes $\csc^{2}\theta$. The expression is now $\frac{\sec^{2}\theta}{\csc^{2}\theta}$.

Step2: Rewrite secant and cosecant in terms of sine and cosine

We know that $\sec\theta=\frac{1}{\cos\theta}$ and $\csc\theta=\frac{1}{\sin\theta}$. So $\sec^{2}\theta=\frac{1}{\cos^{2}\theta}$ and $\csc^{2}\theta=\frac{1}{\sin^{2}\theta}$. Substituting these in, we get $\frac{\frac{1}{\cos^{2}\theta}}{\frac{1}{\sin^{2}\theta}}$.

Step3: Divide the fractions

Dividing by a fraction is multiplying by its reciprocal, so $\frac{1}{\cos^{2}\theta}\times\frac{\sin^{2}\theta}{1}=\frac{\sin^{2}\theta}{\cos^{2}\theta}$.

Step4: Recall the definition of tangent

Since $\tan\theta = \frac{\sin\theta}{\cos\theta}$, then $\tan^{2}\theta=\frac{\sin^{2}\theta}{\cos^{2}\theta}$. So the simplified expression is $\tan^{2}\theta$.

Answer:

b. $\tan^{2}\theta$