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Question: Simplify the following expression: \(\sin {85^\circ} - \sin {35^\circ} - \cos {65^\circ}\) A. 0 ...

Simplify the following expression: sin⁡85∘−sin⁡35∘−cos⁡65∘\sin {85^\circ} - \sin {35^\circ} - \cos {65^\circ}
A. 0
B. 1
C. 2
D. 3

Explanation

Solution

According to give in the question we have to simplify the given trigonometric expression sin⁡85∘−sin⁡35∘−cos⁡65∘\sin {85^\circ} - \sin {35^\circ} - \cos {65^\circ}so, first of all we have to solve the first two terms which are sin⁡85∘\sin {85^\circ}and sin⁡35∘\sin {35^\circ}with the help of the formula as given below:

Formula used: ⇒sin⁡A−sin⁡B=2cos⁡(A+B2)sin⁡(A−B2)....................(1) \Rightarrow \sin A - \sin B = 2\cos \left( {\dfrac{{A + B}}{2}} \right)\sin \left( {\dfrac{{A - B}}{2}} \right)....................(1)
After applying the formula we will have to convert the given term cos⁡65∘\cos {65^\circ} into sin⁡25∘\sin {25^\circ} with the help of the formula as given below:
cos⁡(90∘−θ)=sin⁡θ...................(2)\cos ({90^\circ} - \theta ) = \sin \theta ...................(2)
So with the help of the formula (2) we can convert cos⁡65∘\cos {65^\circ} into sin⁡25∘\sin {25^\circ} and by
eliminating both of the terms obtained in the expression we can simplify it.

Complete step-by-step answer:
Step 1: First of all we will solve the first two terms which are sin⁡85∘\sin {85^\circ} and sin⁡35∘\sin {35^\circ} with the
help of the formula (1) as mentioned in the solution hint.
=(sin⁡85∘−sin⁡35∘)−cos⁡65∘ =2(cos⁡(85∘+35∘2)sin⁡(85∘−35∘2))−cos⁡65∘  = (\sin {85^\circ} - \sin {35^\circ}) - \cos {65^\circ} \\\ = 2\left( {\cos \left( {\dfrac{{{{85}^\circ} + {{35}^\circ}}}{2}} \right)\sin \left( {\dfrac{{{{85}^\circ} - {{35}^\circ}}}{2}} \right)} \right) - \cos {65^\circ} \\\
On solving the expression obtained just above,
=2(cos⁡(120∘2)sin⁡(50∘2))−cos⁡65∘ =2(cos⁡60∘sin⁡25∘)−cos⁡65∘  = 2\left( {\cos \left( {\dfrac{{{{120}^\circ}}}{2}} \right)\sin \left( {\dfrac{{{{50}^\circ}}}{2}} \right)} \right) -\cos {65^\circ} \\\ = 2\left( {\cos {{60}^\circ}\sin {{25}^\circ}} \right) - \cos {65^\circ} \\\
Step 2: Now, to solve the obtained trigonometric expression in step 2 we have to place the value of
cos⁡60∘\cos {60^\circ} and as we know that cos⁡60∘=12\cos {60^\circ} = \dfrac{1}{2}
Hence,
=2(12sin⁡25∘)−cos⁡65∘ =sin⁡25∘−cos⁡65∘  = 2\left( {\dfrac{1}{2}\sin {{25}^\circ}} \right) - \cos {65^\circ} \\\ = \sin {25^\circ} - \cos {65^\circ} \\\
Step 3: Now, we have to convert the given term cos⁡65∘\cos {65^\circ} into sin⁡25∘\sin {25^\circ} with the help of the formula (2) as mentioned in the solution hint.
=sin⁡25∘−cos⁡65∘ =sin⁡25∘−cos⁡(90∘−25∘) =sin⁡25∘−sin⁡25∘ =0  = \sin {25^\circ} - \cos {65^\circ} \\\ = \sin {25^\circ} - \cos ({90^\circ} - {25^\circ}) \\\ = \sin {25^\circ} - \sin {25^\circ} \\\ = 0 \\\
Final solution: Hence, with the help of the formula (1) and (2) we have simplified the given
trigonometric expression sin⁡85∘−sin⁡35∘−cos⁡65∘\sin {85^\circ} - \sin {35^\circ} - \cos {65^\circ}= 0.

Therefore option (A) is correct.

Note: It is necessary to solve the first two terms which are sin⁡85∘\sin {85^\circ} and sin⁡35∘\sin {35^\circ} first to obtain the solution easily of the given expression.
We can convert sin⁡θ\sin \theta to cos⁡θ\cos \theta and cos⁡θ\cos \theta to sin⁡θ\sin \theta with the help of the formula sin⁡(90∘−θ)=cos⁡θ\sin ({90^\circ} - \theta ) = \cos \theta and cos⁡(90∘−θ)=sin⁡θ\cos ({90^\circ} - \theta ) = \sin \theta