show that sec theta(cosec theta) - cot theta = tan theta
secθ cscθ - cotθ
secθ cscθ - cosθ cscθ
(secθ-cosθ) cscθ
(1-cos^2θ)/cosθ cscθ
sin^2θ/cosθ cscθ
sinθ/cosθ
tanθ
To prove the given equation: sec(theta) * cosec(theta) - cot(theta) = tan(theta), we need to use trigonometric identities. Let's break it down step by step:
1. Start with the left-hand side of the equation:
sec(theta) * cosec(theta) - cot(theta)
2. Recall the definitions of the trigonometric functions:
sec(theta) = 1 / cos(theta)
cosec(theta) = 1 / sin(theta)
cot(theta) = cos(theta) / sin(theta)
3. Substitute the above definitions into the left-hand side of the equation:
(1 / cos(theta)) * (1 / sin(theta)) - (cos(theta) / sin(theta))
4. Combine the fractions on the left side (LHS):
(1 * 1) / (cos(theta) * sin(theta)) - (cos(theta) / sin(theta))
5. Find a common denominator for the fractions on the left side (LHS):
(1 * sin(theta)) / (cos(theta) * sin(theta)) - (cos(theta) * cos(theta)) / (sin(theta) * cos(theta))
6. Simplify the expression:
sin(theta) / (cos(theta) * sin(theta)) - (cos^2(theta)) / (sin(theta) * cos(theta))
7. Cancel out the common factors in the numerator and denominator:
1 / cos(theta) - cos(theta) / sin(theta)
8. Convert 1 / cos(theta) to sec(theta) and cos(theta) / sin(theta) to cot(theta):
sec(theta) - cot(theta)
9. Finally, compare this result with the right-hand side (RHS) of the equation:
sec(theta) - cot(theta) = tan(theta)
Therefore, we have shown that sec(theta) * cosec(theta) - cot(theta) = tan(theta) is true by simplifying the left-hand side (LHS) and demonstrating that it is equal to the right-hand side (RHS).