Question

How do you evaluate the exact value of \[\cos 70\cos 20 - \sin 70\sin 20\] ?

Answer 1

Hint:Here in this question to find the exact solution of a given trigonometric function by using the formula of cosine addition rule defined as Cos(A+B) = Cos A. Cos B - Sin A. Sin B where A and B are the angles then by using the value of specified angle of trigonometric ratios on
simplification, we get the required result

Complete step by step answer:
To evaluate the given question by using a formula of cosine addition defined as the cosine addition
formula calculates the cosine of an angle that is either the sum or difference of two other angles. It
arises from the law of cosines and the distance formula. By using the cosine addition formula, the
cosine of both the sum and difference of two angles can be found with the two angles#39; sines and cosines.

i.e., cos(A+B) = cos A. Cos B – sin A. Sin B

Consider the given equation
\[ \Rightarrow \,\,\,\,\cos 70\cos 20 - \sin 70\sin 20\]

Compare the given equation to the formula where A=\[{70^ \circ }\]and B=\[{20^ \circ }\]

Then by formula we can written as
\[\cos 70\cos 20 - \sin 70\sin 20 = \cos (70 + 20)\]

\[ \Rightarrow \,\,\cos \left( {70 + 20} \right)\]

\[ \Rightarrow \,\,\cos {90^ \circ } = 1\]
By the value specified angle of \[\cos {90^ \circ } = 1\]

Hence, the exact value of \[\cos 70\cos 20 - \sin 70\sin 20\] by using the additional cosine formula is 1.

Note: In trigonometry we have trigonometry ratios they are sine, cosine, tangent, cosecant, secant and cotangent. When we have to find the value of trigonometry, we use the table of trigonometry ratios for standard angles. here the given question is in the form of Cos(A+B) = Cos A. Cos B - Sin A. Sin B where A and B are the angles. Hence we can solve these kinds of problems like this method.