Simplify csc(x)^2tan(x)^2-1

Problem

csc(x)*tan(x)−1

Solution

1. Rewrite the trigonometric functions in terms of sine and cosine using the identities csc(x)=1/sin(x) and tan(x)=sin(x)/cos(x)

csc(x)*tan(x)−1=(1/sin(x))2*(sin(x)/cos(x))2−1

2. Simplify the product by canceling the common sin(x) term in the numerator and denominator.

1/sin(x)⋅sin(x)/cos(x)−1=1/cos(x)−1

3. Apply the reciprocal identity sec(x)=1/cos(x) to rewrite the fraction.

1/cos(x)−1=sec(x)−1

4. Use the Pythagorean identity tan(x)+1=sec(x) which implies sec(x)−1=tan(x)

sec(x)−1=tan(x)

Final Answer

csc(x)*tan(x)−1=tan(x)

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