If cosec ? - sin ? = l and sec ? - cos ? = m, prove that l�m�(l� + m� + 3) = 1.
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Step-by-Step Solution
Step 1: Simplify expressions for l and m
First, we express l and m in terms of sinθ and cosθ. We use the identities cosec θ=sinθ1 and secθ=cosθ1. Then, we simplify the expressions using the Pythagorean identity sin2θ+cos2θ=1.
Step 2: Calculate l2 and m2
Next, we square the simplified expressions for l and m to find l2 and m2. This will help us substitute these values into the expression we need to prove.
Step 3: Calculate l2m2
Now, we multiply l2 and m2. Notice that several terms cancel out, simplifying the expression significantly.
Step 4: Calculate l2+m2
Next, we find the sum l2+m2. We combine the fractions by finding a common denominator, which is sin2θcos2θ.
Step 5: Simplify l2+m2 further
We use the sum of cubes factorization a3+b3=(a+b)(a2−ab+b2) where a=cos2θ and b=sin2θ. We also use the identity a2+b2=(a+b)2−2ab to simplify the numerator further. Since sin2θ+cos2θ=1, the expression simplifies to 1−3cos2θsin2θ in the numerator.
Step 6: Substitute into the main expression and simplify
Finally, we substitute the expressions for l2m2 and l2+m2 into the left-hand side of the equation we need to prove. We then simplify the expression by combining the terms inside the parenthesis and cancelling out common factors. This results in 1, which matches the right-hand side of the equation.