Simplify (tan(-x)csc(-x))/(sec(-x)cot(-x))

Problem

(tan(−x)*csc(−x))/(sec(−x)*cot(−x))

Solution

1. Apply even and odd identities to simplify the negative arguments. Recall that tan(−x)=−tan(x) csc(−x)=−csc(x) sec(−x)=sec(x) and cot(−x)=−cot(x)

((−tan(x))*(−csc(x)))/((sec(x))*(−cot(x)))

2. Simplify the signs in the numerator and denominator. The two negatives in the numerator become positive, while the single negative in the denominator remains.

(tan(x)*csc(x))/(−sec(x)*cot(x))

3. Rewrite in terms of sine and cosine to simplify the trigonometric functions. Substitute tan(x)=sin(x)/cos(x) csc(x)=1/sin(x) sec(x)=1/cos(x) and cot(x)=cos(x)/sin(x)

(sin(x)/cos(x)⋅1/sin(x))/(−1/cos(x)⋅cos(x)/sin(x))

4. Cancel common factors within the numerator and the denominator. In the numerator, sin(x) cancels out. In the denominator, cos(x) cancels out.

1/cos(x)/(−1/sin(x))

5. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator.

1/cos(x)⋅(−sin(x))

6. Identify the final trigonometric ratio using the identity tan(x)=sin(x)/cos(x)

−sin(x)/cos(x)=−tan(x)

Final Answer

(tan(−x)*csc(−x))/(sec(−x)*cot(−x))=−tan(x)

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