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Q.

In a quadrilateral ABCD, cosA cosB + sinC sinD = ____.


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Detailed Solution


The sum of all the angles of a quadrilateral is 360°.
So, A+B+C+D=360°
A+B=360°−(C+D)
By taking cosine on the both sides of the equation, we get
cos(A+B)=cos(360°−(C+D))
But we know that, cos(360∘−θ) = cos(θ)
So, we can say that,  cos(A+B) = cos(C+D)
Using the formula, we can simplify it following way:
⇒cosAcosB−sinAsinB = cosC cosB−sinC sinD
Taking the negative parts to the opposite side of the equation, we get
⇒cosAcosB+sinCsinD = cosC cosB+sinAsinB
∴  Option 1 is the correct choice.
 
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