The given expression is:
(sinθ+1)(tanθ−secθ)
Let’s simplify it step by step.
Step 1: Expand the product
Expand the two terms using distributive property:
(sinθ+1)(tanθ−secθ)=(sinθ)(tanθ)−(sinθ)(secθ)+(1)(tanθ)−(1)(secθ)
Simplify each term:
=sinθ⋅tanθ−sinθ⋅secθ+tanθ−secθ
Step 2: Substitute trigonometric identities
- Recall that tanθ=cosθsinθ and secθ=cosθ1.
Substitute these into the expression:
=sinθ⋅cosθsinθ−sinθ⋅cosθ1+cosθsinθ−cosθ1
Step 3: Simplify each term
- The first term becomes cosθsin2θ.
- The second term becomes −cosθsinθ.
- The third term remains cosθsinθ.
- The fourth term remains −cosθ1.
Combine all terms:
=cosθsin2θ−cosθsinθ+cosθsinθ−cosθ1
Step 4: Combine like terms
The −cosθsinθ and +cosθsinθ cancel each other out:
=cosθsin2θ−cosθ1
Factor out cosθ1:
=cosθ1(sin2θ−1)
Step 5: Simplify using Pythagorean identity
Recall that sin2θ−1=−cos2θ. Substitute this:
=cosθ1(−cos2θ)
Simplify:
=−cosθ
Final Answer:
(sinθ+1)(tanθ−secθ)=−cosθ
Let me know if you want more details or have questions!
Here are 5 related questions you can explore:
- What are the key trigonometric identities used to simplify expressions?
- How does the distributive property help in expanding products?
- What is the significance of the Pythagorean identity in trigonometry?
- How can tanθ and secθ be expressed in terms of sinθ and cosθ?
- 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!