Math Problem Statement

(sin θ + 1)(tan θ - sec θ)

Solution

The given expression is:

(sin⁡θ+1)(tan⁡θ−sec⁡θ)(\sin \theta + 1)(\tan \theta - \sec \theta)

Let’s simplify it step by step.

Step 1: Expand the product

Expand the two terms using distributive property:

(sin⁡θ+1)(tan⁡θ−sec⁡θ)=(sin⁡θ)(tan⁡θ)−(sin⁡θ)(sec⁡θ)+(1)(tan⁡θ)−(1)(sec⁡θ)(\sin \theta + 1)(\tan \theta - \sec \theta) = (\sin \theta)(\tan \theta) - (\sin \theta)(\sec \theta) + (1)(\tan \theta) - (1)(\sec \theta)

Simplify each term:

=sin⁡θ⋅tan⁡θ−sin⁡θ⋅sec⁡θ+tan⁡θ−sec⁡θ= \sin \theta \cdot \tan \theta - \sin \theta \cdot \sec \theta + \tan \theta - \sec \theta

Step 2: Substitute trigonometric identities

  • Recall that tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} and sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}.

Substitute these into the expression:

=sin⁡θ⋅sin⁡θcos⁡θ−sin⁡θ⋅1cos⁡θ+sin⁡θcos⁡θ−1cos⁡θ= \sin \theta \cdot \frac{\sin \theta}{\cos \theta} - \sin \theta \cdot \frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta} - \frac{1}{\cos \theta}

Step 3: Simplify each term

  • The first term becomes sin⁡2θcos⁡θ\frac{\sin^2 \theta}{\cos \theta}.
  • The second term becomes −sin⁡θcos⁡θ-\frac{\sin \theta}{\cos \theta}.
  • The third term remains sin⁡θcos⁡θ\frac{\sin \theta}{\cos \theta}.
  • The fourth term remains −1cos⁡θ-\frac{1}{\cos \theta}.

Combine all terms:

=sin⁡2θcos⁡θ−sin⁡θcos⁡θ+sin⁡θcos⁡θ−1cos⁡θ= \frac{\sin^2 \theta}{\cos \theta} - \frac{\sin \theta}{\cos \theta} + \frac{\sin \theta}{\cos \theta} - \frac{1}{\cos \theta}

Step 4: Combine like terms

The −sin⁡θcos⁡θ-\frac{\sin \theta}{\cos \theta} and +sin⁡θcos⁡θ+\frac{\sin \theta}{\cos \theta} cancel each other out:

=sin⁡2θcos⁡θ−1cos⁡θ= \frac{\sin^2 \theta}{\cos \theta} - \frac{1}{\cos \theta}

Factor out 1cos⁡θ\frac{1}{\cos \theta}:

=1cos⁡θ(sin⁡2θ−1)= \frac{1}{\cos \theta} (\sin^2 \theta - 1)

Step 5: Simplify using Pythagorean identity

Recall that sin⁡2θ−1=−cos⁡2θ\sin^2 \theta - 1 = -\cos^2 \theta. Substitute this:

=1cos⁡θ(−cos⁡2θ)= \frac{1}{\cos \theta} (-\cos^2 \theta)

Simplify:

=−cos⁡θ= -\cos \theta

Final Answer:

(sin⁡θ+1)(tan⁡θ−sec⁡θ)=−cos⁡θ(\sin \theta + 1)(\tan \theta - \sec \theta) = -\cos \theta

Let me know if you want more details or have questions!

Here are 5 related questions you can explore:

  1. What are the key trigonometric identities used to simplify expressions?
  2. How does the distributive property help in expanding products?
  3. What is the significance of the Pythagorean identity in trigonometry?
  4. How can tan⁡θ\tan \theta and sec⁡θ\sec \theta be expressed in terms of sin⁡θ\sin \theta and cos⁡θ\cos \theta?
  5. How can this simplification be visualized on a unit circle?

Tip: Always simplify expressions step-by-step, applying one identity at a time to avoid confusion!

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Math Problem Analysis

Mathematical Concepts

Trigonometric identities
Simplification of expressions

Formulas

tan θ = sin θ / cos θ
sec θ = 1 / cos θ
sin^2 θ + cos^2 θ = 1

Theorems

Pythagorean trigonometric identity

Suitable Grade Level

Grades 10-12

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