Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store
seo-qna
SearchIcon
banner

Without using trigonometric tables, evaluate the following:
sec37cosec 53+2cot15cot25cot45cot75cot653(sin218+sin272)

Answer
VerifiedVerified
527.1k+ views
like imagedislike image
Hint: Use trigonometric identities to solve this question. Use the formula for conversion of sec and cosec to cos and sin respectively.
 cosec(θ)=1sin(θ)sec(θ)=1cos(θ)

Use the formula of complementary angles in trigonometry like
sin(θ)=cos(90θ)
tan(90θ)=cot(θ) 

Use the formula
sin2(θ)+cos2(θ)=1

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,
cosec(θ)=1sin(θ) ...(i)sec(θ)=1cos(θ) ...(ii)
Using the equation (i) and equation (ii) we get the first term converted as:
sec37cosec 53=1cos371sin 53=sin53cos37 ...(iii) 
We know that
 sin(θ)=cos(90θ)

Now using the above equation in the numerator of right hand side of equation (iii) we get,
sin53cos37=cos(9053)cos37=cos37cos37=1
So,
sec37cosec 53=1 ...(iv)
We know that
tan(90θ)=cot(θ) 

Now using the above equation we get,
cot15=tan(9015)=tan75
Similarly we can get
cot25=tan(9025)=tan65

Also, we know that
cot45=1
Replacing cot15, cot25and cot45 with the expression as found above in the second term we get the second term as:
2cot15cot25cot45cot75cot65=2tan75tan651cot75cot65

Rearranging the terms in right hand side of the equation we get,
2cot15cot25cot45cot75cot65=2tan75cot75tan65cot65 ...(v)

We know that
 tan(θ)=1cot(θ)tan(θ)cot(θ)=1 ...(vi)

Using equation (vi) we get,
tan75cot75=1

Similarly, tan65cot65=1
Using the above expression in the equation (v) we get,
2cot15cot25cot45cot75cot65=2(tan75cot75)(tan65cot65) =2(1)(1) =2
So, we get that
2cot15cot25cot45cot75cot65=2 ...(vii)

Now coming to the last term of the given expression,
We know that
sin(θ)=cos(90θ)
Using the above equation we get,
 sin72=cos(9072) =cos18

Using the above conversion, the third term is converted to:
 3(sin218+sin272)=3(sin218+(sin72)2) =3(sin218+(cos18)2) =3(sin218+cos218) ...(viii)
We know that
sin2(θ)+cos2(θ)=1
Using the above equation we get,
 sin218+cos218=1 ...(ix)

Using equation (ix) in the equation (viii) we get,
3(sin218+sin272)=3(sin218+cos218) =3(1) =3
So, we get
3(sin218+sin272)=3 ...(x)

Now using equation (iii), equation (vii) and equation (x), adding them all we get,
sec37cosec 53+2cot15cot25cot45cot75cot653(sin218+sin272)=1+23sec37cosec 53+2cot15cot25cot45cot75cot653(sin218+sin272)=0
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 cot45 in the second term which makes the solution procedure wrong though this will not affect the answer.