1)Complete the identity
sec Q - 1/sec Q
Does this equal -2tan^2Q?
To complete the identity, we need to manipulate the expression sec Q - 1/sec Q and see if it simplifies to -2tan^2Q.
First, let's express sec Q and tan Q in terms of sine and cosine. The identity sec Q = 1/cos Q tells us that sec Q = 1/(cos Q). Furthermore, the identity tan Q = sin Q / cos Q tells us that tan^2Q = (sin Q / cos Q)^2.
Now, let's substitute these values into the expression:
sec Q - 1/sec Q = 1/(cos Q) - 1/(1/cos Q)
Next, let's simplify:
sec Q - 1/sec Q = 1/(cos Q) - cos Q
To combine the terms with a common denominator, we can multiply the second term, cos Q, by (cos Q)/(cos Q):
sec Q - 1/sec Q = 1/(cos Q) - cos^2Q/(cos Q)
Now, let's find a common denominator for the two terms:
sec Q - 1/sec Q = (1 - cos^2Q)/(cos Q)
We can further simplify the numerator using the trigonometric identity sin^2Q + cos^2Q = 1. By rearranging the identity, we have 1 - cos^2Q = sin^2Q.
Applying this, we get:
sec Q - 1/sec Q = sin^2Q/(cos Q)
Using the identity tan Q = sin Q / cos Q, we can substitute this value:
sec Q - 1/sec Q = (tan^2Q)/(cos Q)
So, to answer the question, the expression sec Q - 1/sec Q simplifies to (tan^2Q)/(cos Q) or tan^2Q/(cos Q).
In conclusion, the expression does not equal -2tan^2Q, but rather tan^2Q/(cos Q).