Prove the trig identity: 1+Sec^2x/Sec^2x = 1 + cos^2x

1. Start with the left-hand side of the equation:

\frac{1+\sec^2 x}{\sec^2 x}

2. Recall that \sec x = \frac{1}{\cos x} , so \sec^2 x = \frac{1}{\cos^2 x} . Therefore, substitute:

\frac{1+\frac{1}{\cos^2 x}}{\frac{1}{\cos^2 x}}

3. Simplify the fraction:

\frac{\cos^2x+1}{\frac{\cos^2\left(x\right)}{\cos^2x}}

4. Multiply by the reciprocal of the denominator:

(\cos^2x+1)

6. Simplify to get:

\cos^2 x + 1

7. Compare with the right-hand side, which is 1 + \cos^2 x . Since \cos^2 x + 1 is equivalent to 1 + \cos^2 x , the trigonometric identity is proved.

Answer:

1 + \cos^2 x