Exercise$a=\frac{\cos^{2}x}{1+\senx}-\frac{\sen^{2}x}{1+\cos x}$
Step-by-step Solution
1
The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors
$L.C.M.=\left(1+\sin\left(x\right)\right)\left(1+\cos\left(x\right)\right)$
2
Obtained the least common multiple (LCM), we place it as the denominator of each fraction, and in the numerator of each fraction we add the factors that we need to complete
$\frac{\cos\left(x\right)^2\left(1+\cos\left(x\right)\right)}{\left(1+\sin\left(x\right)\right)\left(1+\cos\left(x\right)\right)}+\frac{-\sin\left(x\right)^2\left(1+\sin\left(x\right)\right)}{\left(1+\sin\left(x\right)\right)\left(1+\cos\left(x\right)\right)}$
3
Simplify the numerators
$\frac{\cos\left(x\right)^2+\cos\left(x\right)^2\cos\left(x\right)}{\left(1+\sin\left(x\right)\right)\left(1+\cos\left(x\right)\right)}+\frac{-\sin\left(x\right)^2-\sin\left(x\right)^2\sin\left(x\right)}{\left(1+\sin\left(x\right)\right)\left(1+\cos\left(x\right)\right)}$
Intermediate steps
4
Combine and simplify all terms in the same fraction with common denominator $\left(1+\sin\left(x\right)\right)\left(1+\cos\left(x\right)\right)$
$a=\frac{\cos\left(x\right)^2+\cos\left(x\right)^{3}-\sin\left(x\right)^2-\sin\left(x\right)^{3}}{\left(1+\sin\left(x\right)\right)\left(1+\cos\left(x\right)\right)}$
Explain this step further
Final answer to the exercise $a=\frac{\cos\left(x\right)^2+\cos\left(x\right)^{3}-\sin\left(x\right)^2-\sin\left(x\right)^{3}}{\left(1+\sin\left(x\right)\right)\left(1+\cos\left(x\right)\right)}$