
What is the formula of ?
Answer
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Hint: Here in this question, we have to find the formula of given trigonometric function. To solve this, First rewrite the given angle in the form of addition or difference, then the standard trigonometric formula cosine sum i.e., or cosine difference i.e., identity defined as and using one of these we get required value.
Complete step-by-step solution:
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' sines and cosines.
Consider the given function
-------(1)
It can be rewritten as
Now, by using a cosine sum identity:
Here, and on substituting, we have
As by the standard trigonometric identity: , then on simplification we have
On simplification, we have
Or it can be written as
Again, by the cosine sum identity we have
Again, by the trigonometric identity , we have
Now using a algebraic identity
Here and , then
On simplification, we have
Hence, the formula .
Note: This question also can be solved by using a double angle formula and further simplification will be the same in the above method. Since they have mentioned to solve the given function by using the cosine sum or difference identity, for this we have standard formula and also known the trigonometric identities, double and half angle formulas.
Complete step-by-step solution:
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' sines and cosines.
Consider the given function
It can be rewritten as
Now, by using a cosine sum identity:
Here,
As by the standard trigonometric identity:
On simplification, we have
Or it can be written as
Again, by the cosine sum identity we have
Again, by the trigonometric identity
Now using a algebraic identity
Here
On simplification, we have
Hence, the formula
Note: This question also can be solved by using a double angle formula
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