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Question: How do you find the exact functional value \( \sin {110^o}\sin {70^o} - \cos {110^o}\cos {70^o} \) u...

How do you find the exact functional value sin⁡110osin⁡70o−cos⁡110ocos⁡70o\sin {110^o}\sin {70^o} - \cos {110^o}\cos {70^o} using the cosine sum or difference identity?

Explanation

Solution

Hint : We have to find the value of the given trigonometric expression by first simplifying the expression using the cosine sum or difference identity. The identities are as follows,
cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B   \cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B \\\ \cos \left( {A - B} \right) = \cos A\cos B + \sin A\sin B \;
After simplification we can find the exact value by finding the cosine of the resulting angle.

Complete step-by-step answer :
We have to find the value of the trigonometric function sin⁡110osin⁡70o−cos⁡110ocos⁡70o\sin {110^o}\sin {70^o} - \cos {110^o}\cos {70^o} . For this we have to first simplify the given expression using the cosine sum or difference identity.
The cosine sum identity is given as,
cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B
And the cosine difference identity is given as,
cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos \left( {A - B} \right) = \cos A\cos B + \sin A\sin B
We can observe that the arithmetic sign of both the terms in the given expression is different, so we will use the cosine sum identity.
We can write the given expression as,
sin⁡110osin⁡70o−cos⁡110ocos⁡70o=−(cos⁡110ocos⁡70o−sin⁡110osin⁡70o)\sin {110^o}\sin {70^o} - \cos {110^o}\cos {70^o} = - \left( {\cos {{110}^o}\cos {{70}^o} - \sin {{110}^o}\sin {{70}^o}} \right)
Now if we compare it with the identity we can observe that it becomes similar to the identity with A=110oA = {110^o} and B=70oB = {70^o} .
Thus, using the identity we can write,
−(cos⁡110ocos⁡70o−sin⁡110osin⁡70o)=−(cos⁡(110o+70o))=−(cos⁡180o)- \left( {\cos {{110}^o}\cos {{70}^o} - \sin {{110}^o}\sin {{70}^o}} \right) = - \left( {\cos \left( {{{110}^o} + {{70}^o}} \right)} \right) = - \left( {\cos {{180}^o}} \right)
From basic trigonometric values we know that the value of cos⁡180o=−1\cos {180^o} = - 1 .
Thus, the value of −(cos⁡180o)=−1×−1=1- \left( {\cos {{180}^o}} \right) = - 1 \times - 1 = 1
Hence, the value of sin⁡110osin⁡70o−cos⁡110ocos⁡70o\sin {110^o}\sin {70^o} - \cos {110^o}\cos {70^o} is 11 .
Formula Used:
Cosine sum identity: cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B
So, the correct answer is “1”.

Note : As given in the question we have to use either cosine sum or difference identity while solving. Also, if we use the value of sine or cosine of 110o{110^o} and 70o{70^o} we may not have the exact answer as these values will be in decimals. Thus, it is always a good practice to simplify the given expression using identities.