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Question: Show that: \[\cos \left( {{{35}^ \circ } + A} \right) \cdot \cos \left( {{{35}^ \circ } - B} \right)...

Show that: cos(35+A)cos(35B)+sin(35+A)sin(35B)=cos(A+B)\cos \left( {{{35}^ \circ } + A} \right) \cdot \cos \left( {{{35}^ \circ } - B} \right) + \sin \left( {{{35}^ \circ } + A} \right) \cdot \sin \left( {{{35}^ \circ } - B} \right) = \cos \left( {A + B} \right).

Explanation

Solution

Here in this question, we have to prove the given trigonometric function by showing the left hand side is equal to the right hand side (i.e., L.H.S=R.H.SL.H.S = R.H.S). To solve this, we have to consider L.H.S and simplify by using a formula of cosine and sum identity and by further simplification we get the required solution.

Complete step by step solution:
A function of an angle expressed as the ratio of two of the sides of a right triangle that contains that angle; the sine, cosine, tangent, cotangent, secant, or cosecant known as trigonometric function Also called circular function.
Consider the given question:
show that
cos(35+A)cos(35B)+sin(35+A)sin(35B)=cos(A+B)\Rightarrow \,\,\cos \left( {{{35}^ \circ } + A} \right) \cdot \cos \left( {{{35}^ \circ } - B} \right) + \sin \left( {{{35}^ \circ } + A} \right) \cdot \sin \left( {{{35}^ \circ } - B} \right) = \cos \left( {A + B} \right)--------(1)
Consider Left hand side of equation (1)
L.H.S\Rightarrow \,\,L.H.S
cos(35+A)cos(35B)+sin(35+A)sin(35B)\Rightarrow \,\,\cos \left( {{{35}^ \circ } + A} \right) \cdot \cos \left( {{{35}^ \circ } - B} \right) + \sin \left( {{{35}^ \circ } + A} \right) \cdot \sin \left( {{{35}^ \circ } - B} \right)--------(3)
Now, expand each term using a trigonometric formula of sum and difference identity i.e.,
Sine sum identity: sin(A+B)=sinAcosB+cosAsinB\sin \left( {A + B} \right) = \sin A \cdot \cos B + \cos A \cdot \sin B
Sine difference identity: sin(AB)=sinAcosBcosAsinB\sin \left( {A - B} \right) = \sin A \cdot \cos B - \cos A \cdot \sin B
Cosine sum identity: cos(A+B)=cosAcosBsinAsinB\cos \left( {A + B} \right) = \cos A \cdot \cos B - \sin A \cdot \sin B
Cosine difference identity: cos(AB)=cosAcosB+sinAsinB\cos \left( {A - B} \right) = \cos A \cdot \cos B + \sin A \cdot \sin B
On substituting the formulas the equation (2) becomes
(cos(35)cos(A)sin(35)sin(A))(cos(35)cos(B)+sin(35)sin(B))\Rightarrow \,\,\left( {\cos \left( {{{35}^ \circ }} \right) \cdot \cos \left( A \right) - \sin \left( {{{35}^ \circ }} \right) \cdot \sin \left( A \right)} \right) \cdot \left( {\cos \left( {{{35}^ \circ }} \right) \cdot \cos \left( B \right) + \sin \left( {{{35}^ \circ }} \right) \cdot \sin \left( B \right)} \right) +(sin(35)cos(A)+cos(35)sin(A))(sin(35)cos(B)cos(35)sin(B)) + \left( {\sin \left( {{{35}^ \circ }} \right) \cdot \cos \left( A \right) + \cos \left( {{{35}^ \circ }} \right) \cdot \sin \left( A \right)} \right) \cdot \left( {\sin \left( {{{35}^ \circ }} \right) \cdot \cos \left( B \right) - \cos \left( {{{35}^ \circ }} \right) \cdot \sin \left( B \right)} \right)
On multiplication, we have
cos2(35)cos(A)cos(B)+cos(35)sin(35)cos(A)sin(B)\Rightarrow \,\,{\cos ^2}\left( {{{35}^ \circ }} \right) \cdot \cos \left( A \right)\cos \left( B \right) + \cos \left( {{{35}^ \circ }} \right)\sin \left( {{{35}^ \circ }} \right) \cdot \cos \left( A \right)\sin \left( B \right) cos(35)sin(35)sin(A)cos(B)sin2(35)sin(A)sin(B) - \cos \left( {{{35}^ \circ }} \right)\sin \left( {{{35}^ \circ }} \right) \cdot \sin \left( A \right)\cos \left( B \right) - {\sin ^2}\left( {{{35}^ \circ }} \right) \cdot \sin \left( A \right)\sin \left( B \right) +sin(35)2cos(A)cos(B)sin(35)cos(35)cos(A)sin(B) + \sin {\left( {{{35}^ \circ }} \right)^2} \cdot \cos \left( A \right)\cos \left( B \right) - \sin \left( {{{35}^ \circ }} \right)\cos \left( {{{35}^ \circ }} \right) \cdot \cos \left( A \right)\sin \left( B \right) +cos(35)sin(35)sin(A)cos(B)cos2(35)sin(A)sin(B) + \cos \left( {{{35}^ \circ }} \right)\sin \left( {{{35}^ \circ }} \right) \cdot \sin \left( A \right)\cos \left( B \right) - {\cos ^2}\left( {{{35}^ \circ }} \right) \cdot \sin \left( A \right)\sin \left( B \right)
on simplification and rearranging, we have
cos2(35)cos(A)cos(B)cos2(35)sin(A)sin(B)\Rightarrow \,\,{\cos ^2}\left( {{{35}^ \circ }} \right) \cdot \cos \left( A \right)\cos \left( B \right) - {\cos ^2}\left( {{{35}^ \circ }} \right) \cdot \sin \left( A \right)\sin \left( B \right) +sin(35)2cos(A)cos(B)sin2(35)sin(A)sin(B) + \sin {\left( {{{35}^ \circ }} \right)^2} \cdot \cos \left( A \right)\cos \left( B \right) - {\sin ^2}\left( {{{35}^ \circ }} \right) \cdot \sin \left( A \right)\sin \left( B \right)
Take out common terms, then
cos2(35)(cos(A)cos(B)sin(A)sin(B))+sin(35)2(cos(A)cos(B)sin(A)sin(B))\Rightarrow \,\,{\cos ^2}\left( {{{35}^ \circ }} \right) \cdot \left( {\cos \left( A \right)\cos \left( B \right) - \sin \left( A \right)\sin \left( B \right)} \right) + \sin {\left( {{{35}^ \circ }} \right)^2}\left( {\cos \left( A \right)\cos \left( B \right) - \sin \left( A \right)\sin \left( B \right)} \right)
(cos(A)cos(B)sin(A)sin(B))(cos2(35)+sin(35)2)\Rightarrow \,\,\left( {\cos \left( A \right)\cos \left( B \right) - \sin \left( A \right)\sin \left( B \right)} \right)\left( {{{\cos }^2}\left( {{{35}^ \circ }} \right) + \sin {{\left( {{{35}^ \circ }} \right)}^2}} \right)
As we now the trigonometric identity i.e., cos2θ+sin2θ=1{\cos ^2}\theta + {\sin ^2}\theta = 1, then
(cos(A)cos(B)sin(A)sin(B))(1)\Rightarrow \,\,\left( {\cos \left( A \right)\cos \left( B \right) - \sin \left( A \right)\sin \left( B \right)} \right)\left( 1 \right)
by using cosine sum identity, the above equation becomes
cos(A+B)\Rightarrow \,\,\cos \left( {A + B} \right)
R.H.S\Rightarrow \,\,R.H.S
Therefore, L.H.S=R.H.SL.H.S = R.H.S
cos(35+A)cos(35B)+sin(35+A)sin(35B)=cos(A+B)\cos \left( {{{35}^ \circ } + A} \right) \cdot \cos \left( {{{35}^ \circ } - B} \right) + \sin \left( {{{35}^ \circ } + A} \right) \cdot \sin \left( {{{35}^ \circ } - B} \right) = \cos \left( {A + B} \right)
Hence proved.

Note:
When solving trigonometry based questions, we have to know the definitions of ratios and always remember the standard angles and formulas are useful for solving certain integration problems where a cosine and sum identity may make things much simpler to solve. Thus, in math as well as in physics, these formulae are useful to derive many important identities.