Understanding the Expression
To simplify the expression Sin∅/cosec∅-1 + cos∅/1+sec∅, we will break it down into manageable parts.
Step 1: Simplifying the First Term
- The term Sin∅/cosec∅ can be rewritten.
- Since cosec∅ is the reciprocal of Sin∅, we have:
- Sin∅/cosec∅ = Sin∅/(1/Sin∅) = Sin²∅.
Thus, the first term simplifies to Sin²∅ - 1.
Step 2: Simplifying the Second Term
- The second term cos∅/(1+sec∅) can also be simplified.
- Recall that sec∅ is the reciprocal of cos∅:
- cos∅/(1+sec∅) = cos∅/(1 + 1/cos∅) = cos∅/(cos∅ + 1)/cos∅ = cos²∅/(cos∅ + 1).
Thus, the second term becomes cos²∅/(cos∅ + 1).
Step 3: Combining the Terms
- Now, we can combine both simplified terms:
- Sin²∅ - 1 + cos²∅/(cos∅ + 1).
Step 4: Further Simplification
- We know that Sin²∅ + cos²∅ = 1:
- Therefore, Sin²∅ = 1 - cos²∅.
Combining this with our expression gives:
- (1 - cos²∅) - 1 + cos²∅/(cos∅ + 1).
This ultimately leads to:
- -cos²∅ + cos²∅/(cos∅ + 1).
Final Expression
- The final expression can be evaluated further depending on the context or specific values of ∅. The key takeaway is to break down complicated expressions using fundamental identities and simplifications.