Simplifying the Expression ( cos A cos (90^circ - A) - sin A sin (90^circ - A) )

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In this article, we will explore the trigonometric expression ( cos A cos (90^circ - A) - sin A sin (90^circ - A) ) . We will utilize co-function identities to simplify this expression and determine its value. This is a fundamental concept in trigonometry that is frequently used in various mathematical and engineering applications.

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Using Co-function Identities

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Co-function identities are relationships between trigonometric functions of complementary angles. These identities are particularly useful in simplifying complex trigonometric expressions. The two key co-function identities we will use are:

" "" " ( cos (90^circ - A) sin A )" " ( sin (90^circ - A) cos A )" "" "

Substitution of Co-function Identities

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By substituting these identities into the original expression, we can simplify it significantly. Let's start with the given expression:

" "" " ( cos A cos (90^circ - A) - sin A sin (90^circ - A) )" "" "

Substituting the co-function identities, we get:

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( cos A sin A - sin A cos A )

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This further simplifies to:

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( cos A sin A - sin A cos A 0 )

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So, the value of the expression ( cos A cos (90^circ - A) - sin A sin (90^circ - A) ) is:

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0

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Second Approach Using Trigonometric Identities

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We can also simplify the expression using direct trigonometric identities:

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Using the identity ( cos A cos (90^circ - A) - sin A sin (90^circ - A) ) and substituting the co-function identities, we get:

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( cos A cos (90^circ - A) - sin A sin (90^circ - A) cos A cos (90^circ - A) - sin A sin (90^circ - A) )

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This expression simplifies to:

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( cos A sin A - sin A cos A 0 )

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Thus, the value is:

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0

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Using Triangles to Understand Co-function Identities

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Consider a right triangle with an angle A. The complementary angle is ( 90^circ - A ). Let's denote the sides of the triangle as follows:

" "" "Adjacent to A: ( adjA )" "Opposite to A: ( oppA )" "Adjacent to ( 90^circ - A ): ( adj (90^circ - A) )" "Opposite to ( 90^circ - A ): ( opp (90^circ - A) )" "" "

Using the definitions of trigonometric functions in a right triangle, we have:

" "" " ( cos A frac{adjA}{hyp} )" " ( sin (90^circ - A) frac{adjA}{hyp} )" " ( sin A frac{oppA}{hyp} )" " ( cos (90^circ - A) frac{oppA}{hyp} )" "" "

Thus, substituting these values into the original expression:

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( cos A cos (90^circ - A) - sin A sin (90^circ - A) )

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We get:

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( frac{adjA}{hyp} cdot frac{oppA}{hyp} - frac{oppA}{hyp} cdot frac{adjA}{hyp} 0 )

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Which simplifies to:

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0

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Hence, the value of the expression is conclusively proven to be 0.