Question
Question: Prove the following: \(\dfrac{\cos 9x-\cos 5x}{\sin 17x-\sin 3x}=-\dfrac{\sin 2x}{\cos 10x}\)...
Prove the following:
sin17x−sin3xcos9x−cos5x=−cos10xsin2x
Solution
Hint: For solving this question, we will simplify the term on the left-hand side and prove that it is equal to the term on the right-hand side. And we will use trigonometric formulas like cosC−cosD=−2sin(2C+D)sin(2C−D) and sinC−sinD=2cos(2C+D)sin(2C−D) for simplifying the term on the left-hand side. After that, we will easily prove the desired result.
Complete step-by-step answer:
Given:
We have to prove the following equation:
sin17x−sin3xcos9x−cos5x=−cos10xsin2x
Now, we will simplify the term on the left-hand side and prove that it is equal to the term on the right-hand side.
Now, before we proceed we should know the following formulas:
cosC−cosD=−2sin(2C+D)sin(2C−D)...................(1)sinC−sinD=2cos(2C+D)sin(2C−D)......................(2)
Now, we will use the above two formulas to simplify the term on the left-hand side.
On the left-hand side, we have sin17x−sin3xcos9x−cos5x .
Now, we will use the formula from the equation (1) to write cos9x−cos5x=−2sin7xsin2x in the term on the left-hand side. Then,
sin17x−sin3xcos9x−cos5x⇒sin17x−sin3x−2sin(29x+5x)sin(29x−5x)⇒sin17x−sin3x−2sin7xsin2x
Now, we will use the formula from the equation (2) to write sin17x−sin3x=2cos10xsin7x in the above expression. Then,
sin17x−sin3x−2sin7xsin2x⇒2cos(217x+3x)sin(217x−3x)−2sin7xsin2x⇒2cos10xsin7x−2sin7xsin2x⇒sin7xcos10x−sin7xsin2x⇒−cos10xsin2x
Now, from the above result, we conclude that the value of the expression sin17x−sin3xcos9x−cos5x will be equal to the value of the expression −cos10xsin2x . Then,
sin17x−sin3xcos9x−cos5x=−cos10xsin2x
Now, from the above result, we conclude that the term on the left-hand side is equal to the term on the right-hand side.
Thus, sin17x−sin3xcos9x−cos5x=−cos10xsin2x .
Hence, proved.
Note: Here, the student should first understand what we have to prove in the question. After that, we should proceed in a stepwise manner and apply trigonometric formulas like cosC−cosD=−2sin(2C+D)sin(2C−D) and sinC−sinD=2cos(2C+D)sin(2C−D) correctly. Moreover, while simplifying we should be aware of the result and avoid calculation mistakes while solving.