Final answer:
The equation 1 ÷ (1 + tan(a)) = cot(a) ÷ (1 + cot(a)) is proven by expressing tan(a) and cot(a) in terms of sine and cosine and simplifying both sides to find they are equal.
Step-by-step explanation:
To prove the equation 1 ÷ (1 + tan(a)) = cot(a) ÷ (1 + cot(a)), we start by expressing tan(a) and cot(a) in terms of sine and cosine. We know that tan(a) is equal to sin(a)/cos(a) and cot(a) is the reciprocal of tan(a), which means cot(a) is equal to cos(a)/sin(a).
- Start with the left side of the equation:
1 ÷ (1 + tan(a)) = 1 ÷ (1 + sin(a)/cos(a))
By finding a common denominator, we can express this as:
cos(a) ÷ (cos(a) + sin(a)) - To work on the right side of the equation:
cot(a) ÷ (1 + cot(a)) = (cos(a)/sin(a)) ÷ (1 + (cos(a)/sin(a)))
Again, finding a common denominator gives us:
cos(a) ÷ (sin(a) + cos(a))
Now, we see that both sides of the equation have the same expression, thereby proving that:
1 ÷ (1 + tan(a)) = cot(a) ÷ (1 + cot(a)).