Solving Trigonometric Equations: Finding secθ – tanθ

The problem asks us to find the value of $\sec\theta - \tan\theta$ given that $\sec\theta + \tan\theta = 2$. This type of problem often uses a fundamental trigonometric identity relating secant and tangent.

We know the Pythagorean identity involving secant and tangent: $$ \sec^2\theta - \tan^2\theta = 1 $$

This identity is very useful because the left side is a difference of squares, which can be factored. Factoring the left side, we get:

$$ (\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = 1 $$

Now, we are given the value of $\sec\theta + \tan\theta$ in the question. We are told that:

$$ \sec\theta + \tan\theta = 2 $$

We can substitute this given value into the factored identity:

$$ (\sec\theta - \tan\theta)(2) = 1 $$

We want to find the value of $\sec\theta - \tan\theta$. To do this, we can divide both sides of the equation by 2:

$$ \sec\theta - \tan\theta = \frac{1}{2} $$

Converting the fraction to a decimal, we get:

$$ \sec\theta - \tan\theta = 0.5 $$

Thus, the value of $\sec\theta - \tan\theta$ is 0.5.

Revision Table: Key Trigonometric Identities

Here are some fundamental trigonometric identities that are helpful for solving various problems:

• Pythagorean Identities:
  • $\sin^2\theta + \cos^2\theta = 1$
  • $\sec^2\theta - \tan^2\theta = 1$
  • $\csc^2\theta - \cot^2\theta = 1$
• Reciprocal Identities:
  • $\sec\theta = \frac{1}{\cos\theta}$
  • $\csc\theta = \frac{1}{\sin\theta}$
  • $\cot\theta = \frac{1}{\tan\theta}$
• Quotient Identities:
  • $\tan\theta = \frac{\sin\theta}{\cos\theta}$
  • $\cot\theta = \frac{\cos\theta}{\sin\theta}$

Additional Information: The Relationship Between secθ and tanθ

The identity $\sec^2\theta - \tan^2\theta = 1$ highlights a fundamental relationship between the secant and tangent functions for any angle $\theta$ where they are defined. This identity comes directly from dividing the basic Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ by $\cos^2\theta$.

Let's see how:

Start with: $\sin^2\theta + \cos^2\theta = 1$

Divide all terms by $\cos^2\theta$ (assuming $\cos\theta \neq 0$):

$\frac{\sin^2\theta}{\cos^2\theta} + \frac{\cos^2\theta}{\cos^2\theta} = \frac{1}{\cos^2\theta}$

Using the quotient identity $\tan\theta = \frac{\sin\theta}{\cos\theta}$ and the reciprocal identity $\sec\theta = \frac{1}{\cos\theta}$, we get:

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

Which is $\tan^2\theta + 1 = \sec^2\theta$. Rearranging this gives us the identity we used in the problem:

$\sec^2\theta - \tan^2\theta = 1$

This identity is crucial for solving problems involving secant and tangent, especially when sums or differences of these functions are given.