1)tan 5 degrees + tan 25 degrees / 1 tan 5 degrees tan 25 degrees =
sqrt 3 / 3 Am I correct?
2) Complete the identity
tan^2Q - 3sinQtanQsecQ =
-2tan^2Q Correct?
1) To simplify the expression tan 5 degrees + tan 25 degrees / 1 tan 5 degrees tan 25 degrees, we need to first simplify the individual tan terms.
Starting with tan 5 degrees, we can use a calculator or trigonometric table to find its value. tan 5 is approximately 0.0875.
Next, we find the value of tan 25 degrees, which is approximately 0.4663.
Now, we substitute these values back into the original expression:
(0.0875 + 0.4663) / (1 * 0.0875 * 0.4663)
Simplifying further, we have:
0.5538 / 0.0405
Calculating this division, we get approximately 13.6543.
Therefore, the correct answer is not sqrt 3 / 3.
2) To complete the identity tan^2Q - 3sinQtanQsecQ = -2tan^2Q, we need to manipulate the given expression.
Starting with the left-hand side (LHS), we can rewrite secQ as 1 / cosQ:
tan^2Q - 3sinQtanQ(1 / cosQ)
Next, we can apply the identity tanQ = sinQ / cosQ:
(sinQ / cosQ)^2 - 3sinQ (sinQ / cosQ)(1 / cosQ)
Expanding the square of sinQ / cosQ and simplifying, we have:
(sin^2Q / cos^2Q) - 3sin^2Q / cos^2Q
Combining the fractions, we get:
(sin^2Q - 3sin^2Q) / cos^2Q
Now, simplifying the numerator, we have:
(-2sin^2Q) / cos^2Q
Since sin^2Q is equal to 1 - cos^2Q (according to the Pythagorean identity), we can replace sin^2Q in the numerator:
(-2(1 - cos^2Q)) / cos^2Q
Expanding and simplifying further, we get:
(-2 + 2cos^2Q) / cos^2Q
Factoring out a 2, we have:
2(cos^2Q - 1) / cos^2Q
Using the identity cos^2Q = 1 - sin^2Q, we can further simplify:
2((1 - sin^2Q) - 1) / (1 - sin^2Q)
Simplifying the numerator and denominator, we get:
2(-sin^2Q) / -sin^2Q
The -sin^2Q terms cancel out, resulting in:
2
Therefore, the correct answer is 2, not -2tan^2Q.