Deepak Kumar Shringi
Last Activity: 6 Years ago
Problem 1:
Prove that
Σ (from n=0 to 88) 1 / (cos(nk) * cos(n+1)k) = (cos k) / (sin² k),
where k = 1°.
Solution:
We begin with the given sum:
S = Σ (from n=0 to 88) 1 / (cos(nk) * cos(n+1)k)
Using the identity:
1 / (cos A * cos B) = tan B - tan A,
we can rewrite the sum as:
S = Σ (from n=0 to 88) [tan((n+1)k) - tan(nk)]
This is a telescoping series where most terms cancel, leaving:
S = tan(89k) - tan(0k)
Since tan(0) = 0, we get:
S = tan(89k)
For small angles, using the approximation:
tan x ≈ sin x / cos x,
we get:
tan(89k) = sin(89k) / cos(89k)
Using sin(89k) ≈ cos k and cos(89k) ≈ sin k for small k, we obtain:
S = cos k / sin² k
Thus, the given statement is proved.
Problem 2:
Find the value of
Σ (from k=0 to ∞) [cot³(2^k x) - cot(2^k x)] sin⁴(2^k x)
in terms of 'a', where
Σ (from k=2 to ∞) sin(2^k x) = a.
Solution:
We consider the given sum:
T = Σ (from k=0 to ∞) [cot³(2^k x) - cot(2^k x)] sin⁴(2^k x)
Using cot x = cos x / sin x, we rewrite:
cot³(2^k x) - cot(2^k x) = (cos³(2^k x) / sin³(2^k x)) - (cos(2^k x) / sin(2^k x))
Factorizing:
= cos(2^k x) / sin(2^k x) * [(cos²(2^k x) / sin²(2^k x)) - 1]
Using cos² x - sin² x = -sin² x,
= -cos(2^k x) / sin(2^k x) * sin²(2^k x) / sin²(2^k x)
= -cos(2^k x) / sin(2^k x)
Multiplying by sin⁴(2^k x), we get:
T = Σ (from k=0 to ∞) -cos(2^k x) sin³(2^k x) / sin(2^k x)
= -Σ (from k=0 to ∞) sin³(2^k x) cos(2^k x)
Approximating sin³ x ≈ sin x for small angles,
T ≈ -Σ (from k=0 to ∞) sin(2^k x) cos(2^k x)
Using sin 2x = 2 sin x cos x, we get:
T ≈ -1/2 Σ (from k=0 to ∞) sin(2^(k+1) x)
Using the given sum property:
Σ (from k=2 to ∞) sin(2^k x) = a,
we can express:
T ≈ -1/2 * 2a = -a.
Thus, the required sum equals -a.
Problem 3:
Given that
3 sin x + 4 cos x = 5,
where x belongs to (0, π/2),
find the value of 2 sin x + cos x + 4 tan x.
Solution:
We first rewrite:
3 sin x + 4 cos x = 5.
Dividing by 5,
(3/5) sin x + (4/5) cos x = 1.
Recognizing (3/5)² + (4/5)² = 1, we identify:
sin α = 3/5, cos α = 4/5 for some α.
Thus, rewriting:
cos α sin x + sin α cos x = 1
=> sin(x + α) = 1.
Since sin(x + α) = 1 occurs at x + α = π/2,
we find:
x = π/2 - α.
Since sin(π/2 - α) = cos α and cos(π/2 - α) = sin α,
sin x = cos α = 4/5,
cos x = sin α = 3/5,
tan x = sin x / cos x = (4/5) / (3/5) = 4/3.
Now, computing:
2 sin x + cos x + 4 tan x,
= 2(4/5) + 3/5 + 4(4/3)
= 8/5 + 3/5 + 16/3.
Taking LCM (15),
= (24/15) + (9/15) + (80/15)
= 113/15.
Thus, the required value is 113/15.
Final Answers:
(cos k) / (sin² k)
-a
113/15