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

Find the value of
cos(89o)+cos(87o)+cos(85o)+....+cos(85o)+cos(87o)+cos(89o)

Answer
VerifiedVerified
515.1k+ views
like imagedislike image
Hint: First of all, use cos(θ)=cosθ to simplify the equation. Now multiply and divide the equation by sin1o, take it inside the bracket, and use 2cos A sin B = sin (A + B) – sin (A – B). Now, cancel the like terms and use sin90o=1 and sin0o=0 and finally use 1sinθ=cosecθ to get the required answer.

Complete step-by-step solution -
In this question, we have to find the value of
cos(89o)+cos(87o)+cos(85o)+....+cos(85o)+cos(87o)+cos(89o)
Let us consider the expression given in the question in the below format.
E=cos(89o)+cos(87o)+cos(85o)+....+cos(3o)+cos(1o)+cos(1o)+cos(3o)+......+cos(85o)+cos(87o)+cos(89o)
By adding one term from the end and one term from the beginning, the second term from the end and second term from the beginning and keep repeating this, we get,
E=2(cos89o+cos87o+cos85o+cos83o+....+cos5o+cos3o+cos1o)
Now, multiplying sin1o on both numerator and denominator of the above expression, we get,
E=2sin1osin1o(cos1o+cos3o+cos5o+......+cos87o+cos89o)
We can also write the above equation as,
E=1sin1o(2cos1osin1o+2cos3osin1o+2cos5osin1o+......+2cos87osin1o+2cos89osin1o)
We know that, 2 cos A sin B = sin (A + B) – sin (A – B). By using this in the above expression, we get,
E=1sin1o[sin(1o+1o)sin(1o1o)+sin(3o+1o)sin(3o1o)+sin(5o+1o)sin(5o1o)......+sin(87o+1o)sin(87o1o)+sin(89o+1o)sin(89o1o)]
E=1sin1o[sin2osin0o+sin4osin2o+sin6osin4o......+sin88osin86o+sin90osin88o]
By canceling the like terms, we finally get,
E=1sin1o[sin0o+sin90o]
We know that sin90o=1 and sin0o=0. By substituting these values, we get,
E=1sin1o[0+1]
E=1sin1o
We know that 1sinθ=cosecθ.
So, we get,
E=cosec1o
So, we get the value of the given expression as cosec1o.

Note: In this question, first of all, students often make the mistake of writing cos(θ) as cosθ which is wrong because cos(θ)=cosθ. Now apart from multiplying sin1o, students often try to club 89o with 1o,87o with 3o and so on. They are complementary angles but that leads to lengthy solutions or they often get stuck. And always try to convert the product into different formulas to cancel the terms.


Latest Vedantu courses for you
Grade 11 Science PCM | CBSE | SCHOOL | English
CBSE (2025-26)
calendar iconAcademic year 2025-26
language iconENGLISH
book iconUnlimited access till final school exam
tick
School Full course for CBSE students
PhysicsPhysics
ChemistryChemistry
MathsMaths
₹41,848 per year
Select and buy