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?
To solve these questions, we will use trigonometric identities and properties.
1) To simplify the expression tan 5 degrees + tan 25 degrees / (1 - tan 5 degrees tan 25 degrees), we will start by applying the sum formula for tangent:
tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
Using this formula, the expression becomes:
tan(5 degrees + 25 degrees) / (1 - tan(5 degrees)tan(25 degrees))
tan(30 degrees) / (1 - tan(5 degrees)tan(25 degrees))
Now, we can simplify further using the value of tan(30 degrees), which is √3/3:
(√3/3) / (1 - tan(5 degrees)tan(25 degrees))
To proceed, we need to find the values of tan(5 degrees) and tan(25 degrees). We can use a calculator to find these values. After doing so, let's assume the values are a and b, respectively.
(√3/3) / (1 - ab)
So, the answer will be (√3/3) / (1 - ab), where a is the approximation of tan(5 degrees) and b is the approximation of tan(25 degrees).
2) The second question asks us to complete the identity tan^2Q - 3sinQtanQsecQ.
To approach this, we will use various trigonometric identities. Let's start by writing the identity we want to complete:
tan^2Q - 3sinQtanQsecQ
Next, we will use the Pythagorean identity for tangents:
tan^2Q = sec^2Q - 1
By substituting this into the original expression, we get:
(sec^2Q - 1) - 3sinQtanQsecQ
Expanding further, we have:
sec^2Q - 1 - 3sinQtanQsecQ
Now, let's simplify by factoring out the common term of secQ:
sec^2Q - 1 - 3sinQtanQsecQ = secQ(secQ - 1) - 3sinQtanQsecQ
So, the simplified form of the expression tan^2Q - 3sinQtanQsecQ is secQ(secQ - 1) - 3sinQtanQsecQ.
Therefore, the answer is not -2tan^2Q, but rather secQ(secQ - 1) - 3sinQtanQsecQ.