91Ó°ÊÓ

• Sine, \( \sin A \), is the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosine, \( \cos A \), is the ratio of the adjacent side to the hypotenuse.
• Tangent, \( \tan A \), is the ratio of the opposite side to the adjacent side, which simplifies to \( \frac{\sin A}{\cos A} \).
• Cotangent, \( \cot A \), is the reciprocal of tangent: \( \frac{1}{\tan A} = \frac{\cos A}{\sin A} \).
• Secant, \( \sec A \), is the reciprocal of cosine: \( \frac{1}{\cos A} \).
• Cosecant, \( \operatorname{cosec} A \), is the reciprocal of sine: \( \frac{1}{\sin A} \).

• It allows the replacement of \( \sin^2 A \) with \( 1 - \cos^2 A \) or vice versa, depending on the context.
• This flexibility can be particularly helpful, as seen in many problems where complex expressions are encountered.
• Through various identities similar to the Pythagorean identity, relationships between other trigonometric functions can also be derived. For example, the identity shows up in our exercise when simplifying expressions like \( \frac{\sin^2 A + \cos^2 A}{\cos A \sin A} \).

Steps to Simplification

• First, expand the given trigonometric expression by applying trigonometric identities.
• Apply cancellation where possible. For instance, reducing terms such as \( \frac{\sin A \cos A}{\sin A} \) to \( \cos A \).
• Use identities like the Pythagorean identity to further simplify the expression, as seen when replacing \( \sin^2 A + \cos^2 A \) with 1.
• Finally, reorganize terms to match a desired or given form; this was crucial in proving the provided expression \( \sec A + \operatorname{cosec} A \).