Find dy/dx y=(1-sec(x))/(tan(x))

Problem

d()/d(x)(1−sec(x))/tan(x)

Solution

1. Simplify the expression using trigonometric identities before differentiating to make the calculation easier.

2. Rewrite the terms in the numerator and denominator using sine and cosine.

y=(1−1/cos(x))/sin(x)/cos(x)

3. Multiply the numerator and denominator by cos(x) to clear the complex fraction.

y=(cos(x)−1)/sin(x)

4. Split the fraction into two separate terms.

y=cos(x)/sin(x)−1/sin(x)

5. Apply reciprocal and quotient identities to rewrite the expression in terms of basic trigonometric functions.

y=cot(x)−csc(x)

6. Differentiate each term using the standard derivative rules for trigonometric functions.

d(y)/d(x)=d(cot(x))/d(x)−d(csc(x))/d(x)

7. Substitute the known derivatives d(cot(x))/d(x)=−csc2(x) and d(csc(x))/d(x)=−csc(x)*cot(x)

d(y)/d(x)=−csc2(x)−(−csc(x)*cot(x))

8. Simplify the signs in the resulting expression.

d(y)/d(x)=csc(x)*cot(x)−csc2(x)

Final Answer

d(y)/d(x)=csc(x)*cot(x)−csc2(x)

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