Question

Students were asked to simplify the expression the quantity secant theta minus cosine theta end quantity over secant period Two students' work is given.

Answer 1

Trigonometric Identities

Given the expression:

$\frac{\sec \theta-\cos \theta}{\sec \theta}$

Student A decided to separate the expression into two fractions;

$\frac{\sec \theta}{\sec \theta}-\frac{\cos \theta}{\sec \theta}$

Simplified the first term and substituted the secant for its equivalent as the reciprocal of the cosine:

$1-\frac{\cos \theta}{\frac{1}{\cos \theta}}$

Then simplified the division to get:

$1-\cos ^2\theta$

Finally, used the fundamental identity to get:

$\sin ^2\theta$

Student A did great.

Student B decided to work on the numerator, substituting the secant by the reciprocal of the cosine as follows:

$\frac{\frac{1}{\cos\theta}-\cos \theta}{sec\theta}$

Operated in the numerator:

$\frac{\frac{1-\cos ^2\theta}{\cos \theta}}{sec\theta}$

Used the fundamental identity to get the square of the sine, but here he did a mistake. Multiplied cosine by secant and got cosine squared, which is incorrect. He should write the product and then operate;

$\frac{\sin ^2\theta}{\cos \theta\sec \theta}$

The cosine and the secant are reciprocals of each other, so their product is 1, the denominator vanishes:

$\text{sin}^2\theta$

This way, he should have gotten the same result as student A