Question

Simplify the following expression to a single trig function with no fractions.

1-sin ²(x)/sin (x) cos (x)
Please explain the steps you used and clearly state your final answer.

Answer 1

To simplify the expression 1 - sin²(x)/(sin(x)cos(x)), we can use trigonometric identities to rewrite the expression in terms of a single trig function. Using the identity sin²(x) + cos²(x) = 1, we can rewrite sin²(x) as 1 - cos²(x). Canceling out the common factors, the simplified expression is -cos(x) + sec(x).

Step-by-step explanation:

To simplify the expression 1 - sin²(x)/(sin(x)cos(x)), we can use trigonometric identities to rewrite the expression in terms of a single trig function.

Using the identity sin²(x) + cos²(x) = 1, we can rewrite sin²(x) as 1 - cos²(x). Substituting this into the expression, we get: 1 - (1 - cos²(x))/(sin(x)cos(x)).

Next, we can simplify the expression by canceling out the common factors. The sin(x) in the numerator can be canceled out with the sin(x) in the denominator, and the cos(x) in the numerator can be canceled out with the cos(x) in the denominator. This leaves us with 1 - (1 - cos²(x))/cos(x).

Finally, simplifying further, we can distribute the -1 to both terms in the numerator, resulting in -cos²(x)/cos(x) + 1/cos(x). Combining the two terms, we get -cos(x) + sec(x).

Therefore, the simplified expression is -cos(x) + sec(x).