
Prove the given inverse trigonometric expression, .
Answer
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Hint: In this question, we can see that is common in both of the terms present in LHS of the expression. So, we can take as common. Using the property , we can simplify the expression and then transform the inverse cosine function into inverse sine function.
Complete step-by-step solution -
Let us first consider the LHS of the given expression. Now taking as common, we get
In the above expression, we have . So, we need to convert this into a simpler form.
And we also know that the property that, …………(1)
In the above equation (1), taking to the RHS, we get
………….(2)
We are replacing by in the equation (2), we get
……………..(3)
Now, according to the question we have,
…………….(4)
Using equation (3), and putting the value of in equation (4). We get,
………………..(5)
But according to the question, we have the inverse of sine in RHS. So, here we have to convert it into the inverse of sine.
Converting inverse cosine function to inverse sine function.
Let us assume,
……………..(6)
Taking cosine in both of LHS as well as RHS in the equation (6), we get
Now, we have to find .
For that, we also know the identity
Using this formula, we can get the value of .
Now we have, ……………(7)
Taking the inverse of sine in both LHS as well as RHS in equation (7), we get
…………….(8)
From equation (5), we have .
Using equation (6), we can write the equation (5) as .
And by using equation (8) we can write as, ………(9)
So,
Therefore, LHS = RHS.
Hence, proved.
Note: In this question, one can think to find the principal value of inverse sine functions which are provided in LHS and RHS of the given equation, that is and . If we do so, then only we are increasing the complexity and here, we don’t need the principal value of inverse sine function.
Complete step-by-step solution -
Let us first consider the LHS of the given expression. Now taking
In the above expression, we have
And we also know that the property that,
In the above equation (1), taking
We are replacing
Now, according to the question we have,
Using equation (3), and putting the value of
But according to the question, we have the inverse of sine in RHS. So, here we have to convert it into the inverse of sine.
Converting inverse cosine function to inverse sine function.
Let us assume,
Taking cosine in both of LHS as well as RHS in the equation (6), we get
Now, we have to find
For that, we also know the identity
Using this formula, we can get the value of
Now we have,
Taking the inverse of sine in both LHS as well as RHS in equation (7), we get
From equation (5), we have
Using equation (6), we can write the equation (5) as
And by using equation (8) we can write
So,
Therefore, LHS = RHS.
Hence, proved.
Note: In this question, one can think to find the principal value of inverse sine functions which are provided in LHS and RHS of the given equation, that is
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