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How do you prove the identity #cot(x)+sec(x)=(cos(x)+tan(x))/sin(x)#?

Answer 1

Oliver Atkinson

# cot x + sec x = cos x/sin x + 1/cos x = cos x/sin x + sin x/(sin x cos x) = (cos x + tan x)/sin x #

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Answer 2

Anna Andrews

To prove the identity cot(x) + sec(x) = (cos(x) + tan(x))/sin(x), we'll start with the left-hand side (LHS) and manipulate it to match the right-hand side (RHS) of the equation.

LHS: cot(x) + sec(x) = (cos(x)/sin(x)) + (1/cos(x)) [Definition of cotangent and secant] = (cos(x) + sin(x))/sin(x) [Multiplying the second term by sin(x)/sin(x)] = (cos(x)/sin(x)) + (sin(x)/sin(x)) [Breaking up the fraction] = (cos(x) + tan(x))/sin(x) [Definition of tangent]

Thus, we have shown that LHS = RHS, and the identity is proved.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.