Math Problem Statement

sec x - sec x sin ^2 c = cos x

Solution

To solve the given equation, let's rewrite it in a more standard form:

secxsecxsin2x=cosx\sec x - \sec x \sin^2 x = \cos x

Step 1: Factor out secx\sec x on the left side

The left side can be factored as follows:

secx(1sin2x)=cosx\sec x (1 - \sin^2 x) = \cos x

Step 2: Simplify using a trigonometric identity

We know that 1sin2x=cos2x1 - \sin^2 x = \cos^2 x, so we can substitute this into the equation:

secxcos2x=cosx\sec x \cdot \cos^2 x = \cos x

Step 3: Substitute secx=1cosx\sec x = \frac{1}{\cos x}

Now, replace secx\sec x with 1cosx\frac{1}{\cos x}:

cos2xcosx=cosx\frac{\cos^2 x}{\cos x} = \cos x

Simplify the left side:

cosx=cosx\cos x = \cos x

This is an identity, meaning it is true for all values of xx where cosx0\cos x \neq 0.

Solution

The equation holds for all xx where cosx0\cos x \neq 0, or equivalently:

x(2n+1)π2for integer nx \neq \frac{(2n + 1)\pi}{2} \quad \text{for integer } n

Would you like further details or clarification on any step?

Here are some additional questions for practice:

  1. What values of xx satisfy secx=cosx\sec x = \cos x?
  2. How would you solve secx+secxsin2x=cosx\sec x + \sec x \sin^2 x = \cos x?
  3. What are the properties of secx\sec x and how do they relate to cosx\cos x?
  4. Can you prove that 1sin2x=cos2x1 - \sin^2 x = \cos^2 x using the Pythagorean identity?
  5. How do you solve trigonometric equations involving both sec\sec and cos\cos?

Tip: Always look for trigonometric identities that can simplify expressions before solving the equation.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Trigonometric Identities
Equation Solving

Formulas

sec x = 1 / cos x
1 - sin^2 x = cos^2 x

Theorems

Pythagorean Identity

Suitable Grade Level

Grades 10-12