Question

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

Answer 1

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

Formula used: $ \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 $\cos {65^\circ}$ into $\sin
{25^\circ}$ with the help of the formula as given below:
$\cos ({90^\circ} - \theta ) = \sin \theta ...................(2)$
So with the help of the formula (2) we can convert $\cos {65^\circ}$ into $\sin {25^\circ}$ 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 $\sin {85^\circ}$ and $\sin {35^\circ}$ with the
help of the formula (1) as mentioned in the solution hint.
$
= (\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\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
$\cos {60^\circ}$ and as we know that $\cos {60^\circ} = \dfrac{1}{2}$
Hence,
$
= 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 $\cos {65^\circ}$ into $\sin {25^\circ}$ with the help of the formula (2) as mentioned in the solution hint.
$
= \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 $\sin {85^\circ} - \sin {35^\circ} - \cos {65^\circ}$= 0.

Therefore option (A) is correct.

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