Math Problem Statement

Prove the identity: \( \frac{\sec \theta \sin \theta}{\tan \theta + \cot \theta} = \sin^2 \theta \)

Solution

To prove the identity:

sec⁡θsin⁡θtan⁡θ+cot⁡θ=sin⁡2θ\frac{\sec \theta \sin \theta}{\tan \theta + \cot \theta} = \sin^2 \theta

we'll work with each part of the expression separately. Let's proceed with simplifying the left-hand side.

Step 1: Rewrite the Expression

The left side of the equation is:

sec⁡θsin⁡θtan⁡θ+cot⁡θ\frac{\sec \theta \sin \theta}{\tan \theta + \cot \theta}

Recall the definitions of secant, tangent, and cotangent:

  • sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}
  • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}
  • cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}

Step 2: Substitute these Definitions

Substitute sec⁡θ\sec \theta, tan⁡θ\tan \theta, and cot⁡θ\cot \theta into the expression:

1cos⁡θ⋅sin⁡θsin⁡θcos⁡θ+cos⁡θsin⁡θ\frac{\frac{1}{\cos \theta} \cdot \sin \theta}{\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}}

Simplify the numerator:

sin⁡θcos⁡θ\frac{\sin \theta}{\cos \theta}

Now, simplify the denominator by finding a common denominator:

sin⁡2θ+cos⁡2θsin⁡θcos⁡θ\frac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta}

Step 3: Use the Pythagorean Identity

We know that sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1, so the denominator simplifies to:

1sin⁡θcos⁡θ\frac{1}{\sin \theta \cos \theta}

Now we have:

sin⁡θcos⁡θ1sin⁡θcos⁡θ\frac{\frac{\sin \theta}{\cos \theta}}{\frac{1}{\sin \theta \cos \theta}}

Step 4: Simplify the Complex Fraction

Dividing by a fraction is equivalent to multiplying by its reciprocal:

sin⁡θcos⁡θ⋅sin⁡θcos⁡θ1=sin⁡2θ\frac{\sin \theta}{\cos \theta} \cdot \frac{\sin \theta \cos \theta}{1} = \sin^2 \theta

Conclusion

The left-hand side simplifies to sin⁡2θ\sin^2 \theta, which matches the right-hand side. Thus, we have proven the identity:

sec⁡θsin⁡θtan⁡θ+cot⁡θ=sin⁡2θ\frac{\sec \theta \sin \theta}{\tan \theta + \cot \theta} = \sin^2 \theta

Would you like further details on any of the steps, or do you have any questions?

Here are 5 related questions:

  1. How can the Pythagorean identity help simplify trigonometric expressions?
  2. What is the significance of rewriting trigonometric identities using basic functions like sine and cosine?
  3. How do you approach proving a trigonometric identity?
  4. How does understanding reciprocal identities help in simplifying complex trigonometric expressions?
  5. What are some other commonly used trigonometric identities?

Tip: When proving identities, try converting all functions into sines and cosines, as it often makes simplification easier.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Trigonometric Identities
Reciprocal Identities

Formulas

\( \sec \theta = \frac{1}{\cos \theta} \)
\( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
\( \cot \theta = \frac{\cos \theta}{\sin \theta} \)
Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)

Theorems

Pythagorean Identity

Suitable Grade Level

Grades 10-12

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