Answer:
We can start with the left side of the identity and try to manipulate it algebraically to transform it into the right side:
Left side: tan X + cot X
We know that cot X is equal to 1/tan X, so we can substitute this expression in for cot X:
Left side: tan X + 1/tan X
Next, we can use the identity that (a + 1/a) = (a^2 + 1)/a to rewrite the expression as a single fraction:
Left side: (tan^2 X + 1)/tan X
We can then use the trigonometric identity that 1 + tan^2 X = sec^2 X to substitute in for the numerator:
Left side: sec^2 X / tan X
Finally, we can use the identity that csc X = 1/sin X to substitute in for tan X:
Left side: sec^2 X / (sin X / cos X)
We can simplify this expression by multiplying the numerator and denominator by cos X:
Left side: (sec^2 X * cos X) / sin X
We know that sec X is equal to 1/cos X, so we can substitute this expression in for sec X:
Left side: (cos X / cos X * sin X) = cos X * csc X
Therefore, we have shown that the left side (tan X + cot X) is equal to the right side (sec X csc X), and the identity is proven:
tan X + cot X = sec X csc X