Final answer:
To prove (sec x)/(tan x) - (csc x)/(cot x) = 1/sin x - 1/cos x, start with the left-hand side of the equation and simplify it step by step to reach the right-hand side.
Step-by-step explanation:
Begin by expressing secant and cosecant in terms of sine and cosine. Secant x = 1/cos x and cosecant x = 1/sin x. Substitute these expressions into the left-hand side of the equation: (1/cos x)/(tan x) - (1/sin x)/(cot x). Next, rewrite the trigonometric functions in terms of sine and cosine. Tan x = sin x / cos x and cot x = cos x / sin x. Substituting these values into the equation results in: (1/cos x) / (sin x/cos x) - (1/sin x) / (cos x / sin x).
Further simplify the equation by dividing by a fraction, which is equivalent to multiplying by its reciprocal. This simplifies to: (1/cos x) * (cos x/sin x) - (1/sin x) * (sin x/cos x). Cancel out the terms, leaving 1/sin x - 1/cos x. This matches the right-hand side of the original equation, confirming the equality: 1/sin x - 1/cos x = 1/sin x - 1/cos x.
Through the substitution of secant, cosecant, tangent, and cotangent functions in terms of sine and cosine, the left-hand side is manipulated to match the right-hand side, demonstrating that the original equation is valid. This process showcases how trigonometric identities can be used to simplify and prove trigonometric equations.