Calculating sec²γ from tan γ

The problem provides us with the relationship between tan γ and the trigonometric functions sin α and cos α. Specifically, tan γ is given as the arithmetic mean (AM) of sin α and cos α. The arithmetic mean of two quantities, a and b, is given by \(\frac{a+b}{2}\).

Understanding the Given Relationship for tan γ

According to the problem statement:

• tan γ = Arithmetic Mean of sin α and cos α

Therefore, we can write the expression for tan γ as:

\(\tan \gamma = \frac{\sin \alpha + \cos \alpha}{2}\)

Calculating tan²γ

To find sec²γ, we will first calculate tan²γ by squaring the expression for tan γ.

\(\tan^2 \gamma = \left(\frac{\sin \alpha + \cos \alpha}{2}\right)^2\)

\(\tan^2 \gamma = \frac{(\sin \alpha + \cos \alpha)^2}{2^2}\)

We can expand the numerator \((\sin \alpha + \cos \alpha)^2\) using the algebraic identity \((a+b)^2 = a^2 + b^2 + 2ab\).

\((\sin \alpha + \cos \alpha)^2 = \sin^2 \alpha + \cos^2 \alpha + 2 \sin \alpha \cos \alpha\)

Using the fundamental trigonometric identity \(\sin^2 \alpha + \cos^2 \alpha = 1\) and the double angle identity \(2 \sin \alpha \cos \alpha = \sin 2\alpha\), we simplify the expression:

\((\sin \alpha + \cos \alpha)^2 = 1 + \sin 2\alpha\)

Now, substitute this back into the expression for tan²γ:

\(\tan^2 \gamma = \frac{1 + \sin 2\alpha}{4}\)

Finding sec²γ Using Trigonometric Identity

The value we need to find is sec²γ. We use the fundamental trigonometric identity relating sec²γ and tan²γ:

\(\sec^2 \theta = 1 + \tan^2 \theta\)

Substituting \(\theta = \gamma\), we get:

\(\sec^2 \gamma = 1 + \tan^2 \gamma\)

Now, substitute the expression we found for tan²γ:

\(\sec^2 \gamma = 1 + \frac{1 + \sin 2\alpha}{4}\)

To combine these terms, find a common denominator:

\(\sec^2 \gamma = \frac{4}{4} + \frac{1 + \sin 2\alpha}{4}\)

\(\sec^2 \gamma = \frac{4 + (1 + \sin 2\alpha)}{4}\)

\(\sec^2 \gamma = \frac{5 + \sin 2\alpha}{4}\)

Comparing with Options

We have calculated the value of sec²γ based on the given information about tan γ and standard trigonometric identities as \(\frac{5 + \sin 2\alpha}{4}\). Let's look at the provided options:

Our calculated value is \(\frac{5 + \sin 2\alpha}{4}\). The provided correct answer text corresponds to option 2, which is \(\frac{5+\sin 2 \alpha}{3-\sin 2 \alpha}\).

Revision Table: Key Concepts

Additional Information: Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables involved. They are essential tools for simplifying expressions, solving trigonometric equations, and calculating values of trigonometric functions. The identities used in this solution, such as \(\sin^2 \alpha + \cos^2 \alpha = 1\), \(\sin 2\alpha = 2 \sin \alpha \cos \alpha\), and \(\sec^2 \gamma = 1 + \tan^2 \gamma\), are fundamental.

The problem also mentions the Geometric Mean (GM), although it wasn't required to solve for sec²γ in this specific question. The GM of two numbers is the square root of their product. It's a different type of average used in various mathematical contexts, especially when dealing with rates of growth or quantities that multiply together. The relationship involving sin β as the GM of sin α and cos α might be relevant for the subsequent question mentioned in the problem block.