What is sec 2x – tan 2x in terms of tan ?

To express the equation sec 2x – tan 2x in terms of tan, we can use some fundamental trigonometric identities. Let's break it down step by step.

Understanding the Trigonometric Functions

First, recall the definitions of secant and tangent in terms of sine and cosine:

• Secant: sec θ = 1/cos θ
• Tangent: tan θ = sin θ/cos θ

Applying the Double Angle Formulas

Next, we need to use the double angle formulas for cosine and sine:

• cos(2x) = cos²(x) - sin²(x)
• sin(2x) = 2sin(x)cos(x)

Rewriting sec 2x and tan 2x

Now, we can rewrite sec 2x and tan 2x using these identities:

• sec(2x) = 1/cos(2x) = 1/(cos²(x) - sin²(x))
• tan(2x) = sin(2x)/cos(2x) = (2sin(x)cos(x))/(cos²(x) - sin²(x))

Combining the Terms

Now, substituting these into the original expression sec 2x – tan 2x gives us:

sec(2x) - tan(2x) = 1/(cos²(x) - sin²(x)) - (2sin(x)cos(x))/(cos²(x) - sin²(x))

Finding a Common Denominator

Since both terms share the same denominator, we can combine them:

sec(2x) - tan(2x) = (1 - 2sin(x)cos(x)) / (cos²(x) - sin²(x))

Expressing in Terms of Tan

To express this in terms of tan, we can use the identity tan(x) = sin(x)/cos(x). Thus, we can rewrite sin(x) and cos(x) in terms of tan:

• sin(x) = tan(x)cos(x)
• cos(x) = cos(x)

Substituting these back into our expression can get complicated, but we can simplify it further by recognizing that:

1 - 2sin(x)cos(x) = 1 - 2(tan(x)cos(x))(cos(x)) = 1 - 2tan(x)cos²(x)

Final Expression

Thus, the expression sec 2x – tan 2x can be simplified to:

sec(2x) - tan(2x) = (1 - 2tan(x)cos²(x)) / (cos²(x) - sin²(x))

In conclusion, while the expression sec 2x – tan 2x can be rewritten in terms of tan, it often involves a combination of identities and careful manipulation. The final form may still contain cos(x) unless you substitute it entirely in terms of tan, which can lead to more complex expressions. Understanding these relationships is crucial for mastering trigonometric identities.