
Without using trigonometric tables, evaluate the following:
Answer
527.1k+ views
Hint: Use trigonometric identities to solve this question. Use the formula for conversion of sec and cosec to cos and sin respectively.
Use the formula of complementary angles in trigonometry like
Use the formula
Complete step-by-step answer:
This is a very basic question on trigonometric identities. In this question many trigonometric identities are to be used.
We know that,
Using the equation (i) and equation (ii) we get the first term converted as:
We know that
Now using the above equation in the numerator of right hand side of equation (iii) we get,
So,
We know that
Now using the above equation we get,
Similarly we can get
Also, we know that
Replacing , and with the expression as found above in the second term we get the second term as:
Rearranging the terms in right hand side of the equation we get,
We know that
Using equation (vi) we get,
Similarly,
Using the above expression in the equation (v) we get,
So, we get that
Now coming to the last term of the given expression,
We know that
Using the above equation we get,
Using the above conversion, the third term is converted to:
We know that
Using the above equation we get,
Using equation (ix) in the equation (viii) we get,
So, we get
Now using equation (iii), equation (vii) and equation (x), adding them all we get,
Hence we get the value of the expression in the question as 0.
Note: This question is totally based on trigonometric identities so you should be well familiar with every identity otherwise you cannot do this question. There is one more way to simplify the second term, you can convert the next two factors into tan and then proceed to make a group of tan and cot to make it 1. There is a chance of omitting in the second term which makes the solution procedure wrong though this will not affect the answer.
Use the formula of complementary angles in trigonometry like
Use the formula
Complete step-by-step answer:
This is a very basic question on trigonometric identities. In this question many trigonometric identities are to be used.
We know that,
Using the equation (i) and equation (ii) we get the first term converted as:
We know that
Now using the above equation in the numerator of right hand side of equation (iii) we get,
So,
We know that
Now using the above equation we get,
Similarly we can get
Also, we know that
Replacing
Rearranging the terms in right hand side of the equation we get,
We know that
Using equation (vi) we get,
Similarly,
Using the above expression in the equation (v) we get,
So, we get that
Now coming to the last term of the given expression,
We know that
Using the above equation we get,
Using the above conversion, the third term is converted to:
We know that
Using the above equation we get,
Using equation (ix) in the equation (viii) we get,
So, we get
Now using equation (iii), equation (vii) and equation (x), adding them all we get,
Hence we get the value of the expression in the question as 0.
Note: This question is totally based on trigonometric identities so you should be well familiar with every identity otherwise you cannot do this question. There is one more way to simplify the second term, you can convert the next two factors into tan and then proceed to make a group of tan and cot to make it 1. There is a chance of omitting
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