Final answer:
To prove the trigonometric identity 1 + tan^2x / csc^2x = tan^2x, we can start by expressing everything in terms of sine and cosine. Then, we can simplify the left side by multiplying the terms and combining like terms. Finally, we can use the trigonometric identity sin^2(x) * cos^2(x) = sin^2(2x) / 4 to rewrite the right side.
Step-by-step explanation:
To prove the trigonometric identity 1 + tan^2x / csc^2x = tan^2x, we can start by expressing everything in terms of sine and cosine. We know that tan(x) = sin(x) / cos(x) and csc(x) = 1 / sin(x). Therefore:
1 + (sin^2(x) / cos^2(x)) / (1 / sin^2(x)) = (sin^2(x) / cos^2(x))
Next, we can simplify the left side by multiplying the terms and combining like terms:
1 + sin^2(x) * sin^2(x) / cos^2(x) = sin^2(x)
To simplify this further, we can rewrite the left side as a common denominator:
(cos^2(x) + sin^2(x) * sin^2(x)) / cos^2(x) = sin^2(x)
Since cos^2(x) + sin^2(x) = 1, we can substitute it into the equation:
(1 + sin^4(x)) / cos^2(x) = sin^2(x)
Multiplying both sides by cos^2(x), we get:
1 + sin^4(x) = sin^2(x) * cos^2(x)
Finally, we can use the trigonometric identity sin^2(x) * cos^2(x) = sin^2(2x) / 4 to rewrite the right side:
1 + sin^4(x) = sin^2(2x) / 4