Explanation
To solve this problem, we use standard trigonometric identities:
sec2θ−tan2θ=1
csc2θ−cot2θ=1
Step 1: Simplify x and y
Given x=secθ−tanθ. We know that (secθ−tanθ)(secθ+tanθ)=sec2θ−tan2θ=1.
Therefore:
secθ+tanθ=x1
Given y=cscθ+cotθ. We know that (cscθ+cotθ)(cscθ−cotθ)=csc2θ−cot2θ=1.
Therefore:
cscθ−cotθ=y1
Step 2: Express sec,tan,csc,cot in terms of x and y
Step 3: Test the options
Let us pick a value for θ to check the options. Let θ=45∘:
x=sec45∘−tan45∘=2−1
y=csc45∘+cot45∘=2+1
Calculate terms:
Now substitute these into the options:
(a) x+y−xy−1=22−1−1=22−2=0
(b) x−y+xy+1=(2−1)−(2+1)+1+1=−2+2=0
Since option (b) satisfies the condition for θ=45∘, it is the correct relation.
Conclusion: The correct option is (b) x−y+xy+1=0.
