Math Problem Statement

(sin θ + 1)(tan θ - sec θ)

Solution

The given expression is:

(sinθ+1)(tanθsecθ)(\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:

(sinθ+1)(tanθsecθ)=(sinθ)(tanθ)(sinθ)(secθ)+(1)(tanθ)(1)(secθ)(\sin \theta + 1)(\tan \theta - \sec \theta) = (\sin \theta)(\tan \theta) - (\sin \theta)(\sec \theta) + (1)(\tan \theta) - (1)(\sec \theta)

Simplify each term:

=sinθtanθsinθsecθ+tanθsecθ= \sin \theta \cdot \tan \theta - \sin \theta \cdot \sec \theta + \tan \theta - \sec \theta


Step 2: Substitute trigonometric identities

  • Recall that tanθ=sinθcosθ\tan \theta = \frac{\sin \theta}{\cos \theta} and secθ=1cosθ\sec \theta = \frac{1}{\cos \theta}.

Substitute these into the expression:

=sinθsinθcosθsinθ1cosθ+sinθcosθ1cosθ= \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 sin2θcosθ\frac{\sin^2 \theta}{\cos \theta}.
  • The second term becomes sinθcosθ-\frac{\sin \theta}{\cos \theta}.
  • The third term remains sinθcosθ\frac{\sin \theta}{\cos \theta}.
  • The fourth term remains 1cosθ-\frac{1}{\cos \theta}.

Combine all terms:

=sin2θcosθsinθcosθ+sinθcosθ1cosθ= \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 sinθcosθ-\frac{\sin \theta}{\cos \theta} and +sinθcosθ+\frac{\sin \theta}{\cos \theta} cancel each other out:

=sin2θcosθ1cosθ= \frac{\sin^2 \theta}{\cos \theta} - \frac{1}{\cos \theta}

Factor out 1cosθ\frac{1}{\cos \theta}:

=1cosθ(sin2θ1)= \frac{1}{\cos \theta} (\sin^2 \theta - 1)


Step 5: Simplify using Pythagorean identity

Recall that sin2θ1=cos2θ\sin^2 \theta - 1 = -\cos^2 \theta. Substitute this:

=1cosθ(cos2θ)= \frac{1}{\cos \theta} (-\cos^2 \theta)

Simplify:

=cosθ= -\cos \theta


Final Answer:

(sinθ+1)(tanθsecθ)=cosθ(\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 tanθ\tan \theta and secθ\sec \theta be expressed in terms of sinθ\sin \theta and cosθ\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