Simplifying the Expression ( cos A cos (90^circ - A) - sin A sin (90^circ - A) )
" "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.
" "Using Co-function Identities
" "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
" "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:
" "( cos A sin A - sin A cos A )
" "This further simplifies to:
" "( cos A sin A - sin A cos A 0 )
" "So, the value of the expression ( cos A cos (90^circ - A) - sin A sin (90^circ - A) ) is:
" "0
" "Second Approach Using Trigonometric Identities
" "We can also simplify the expression using direct trigonometric identities:
" "Using the identity ( cos A cos (90^circ - A) - sin A sin (90^circ - A) ) and substituting the co-function identities, we get:
" "( cos A cos (90^circ - A) - sin A sin (90^circ - A) cos A cos (90^circ - A) - sin A sin (90^circ - A) )
" "This expression simplifies to:
" "( cos A sin A - sin A cos A 0 )
" "Thus, the value is:
" "0
" "Using Triangles to Understand Co-function Identities
" "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:
" "( cos A cos (90^circ - A) - sin A sin (90^circ - A) )
" "We get:
" "( frac{adjA}{hyp} cdot frac{oppA}{hyp} - frac{oppA}{hyp} cdot frac{adjA}{hyp} 0 )
" "Which simplifies to:
" "0
" "Hence, the value of the expression is conclusively proven to be 0.