Differentiate the following w.r.t x: tan^-1(cos x + sin x)/(cos x - sin x)

Question

Differentiate the following w.r.t x: tan^-1\(\left(\cfrac{cos\,\text x+sin\,\text x}{cos\,\text x-sin\,\text x}\right)\)

tan^-1(cos x + sin x)/(cos x - sin x)

Answer 1

To find: Value of tan^-1\(\left(\cfrac{cos\,\text x+sin\,\text x}{cos\,\text x-sin\,\text x}\right)\)

tan^-1(cos x + sin x)/(cos x - sin x)

Formula used: (i) tan(A + B) = \(\cfrac{tanA+tanB}{1-tanAtanB}\)

We have, tan^-1\(\left(\cfrac{cos\,\text x+sin\,\text x}{cos\,\text x-sin\,\text x}\right)\)

Dividing numerator and denominator by cos x

Now, we can see that  tan^-1\(\left(\cfrac{cos\,\text x+sin\,\text x}{cos\,\text x-sin\,\text x}\right)\) = \(\cfrac{n}4+\text x\)

Now differentiating,