# Answer in Trigonometry for umar #250578

Determine the numerical value of the following expression without the use of a calculator:

log10 (1000100)

100

+

X100

n=1

sin(n) + 1

( 1)n

!

vuut

1Y000

m=1

1

cos(m)2

**Question: **Determine the numerical value of the following expression without the use of a calculator:

“(frac{log_{10} (1000^{100})} {100} + sum_{n = 1}^{100}frac{sin(pi n)+1}{(-1)^n}) . sqrt{prod_{m=1}^{1000}frac{1}{cos (pi m)^{2}}}”

**Solution:**

*For the logarithm operation:*

“frac{log_{10} (1000^{100})} {100}”

“= frac{100log_{10} 1000} {100}”

“= frac{100log_{10} 10^3} {100}”

“= frac{(3times 100)log_{10} 10} {100}”

*since* “log_{10} 10 = 1”;

*Then,*

“therefore frac{log_{10} (1000^{100})} {100} = frac{(3times 100)log_{10} 10} {100} = frac{(3times 100 times 1)} {100} = 3”

*For the cycle operations:*

“f(n) = sum_{n = 1}^{100}frac{sin(pi n)+1}{(-1)^n}”

“for all {n in tmathbb{N}}, sin(pi n) = 0 \ considering the denominator, \ for even n, frac{sin(pi n) + 1} {(-1)^n} =1 \ for odd n, frac{sin(pi n) + 1} {(-1)^n} = -1”

Therefore, dividing the summation into two parts: i.e. 50 odds and 50 evens, we have:

“f(n) = \ 50 times (frac{sin(pi n) + 1} {(-1)^n}) for denominator n = evens \ + 50 times (frac{sin(pi n) + 1} {(-1)^n}) for denominator n = odds \ = (50 times 1) + (50times -1) = 50 – 50 = 0”

“therefore f(n) = sum_{n = 1}^{100}frac{sin(pi n)+1}{(-1)^n} = 0”

*Also, for:*

“f(m) = sqrt{prod_{m=1}^{1000}frac{1}{cos (pi m)^{2}}}”

“for all {m in mathbb{N}}, frac{1} {cos(pi m)^2} = 1”

Therefore the product of:

“prod_{m=1}^{1000}frac{1}{cos (pi m)^{2}} = 1 times 1 times 1 times…….. = 1 \n\ therefore the square root of the product of: \ sqrt{prod_{m=1}^{1000}frac{1}{cos (pi m)^{2}}} = sqrt{1} = 1”

“therefore f(m) = sqrt{prod_{m=1}^{1000}frac{1}{cos (pi m)^{2}}} = 1”

*Therefore, the final answer to the expression is given by:*

“therefore (frac{log_{10} (1000^{100})} {100} + sum_{n = 1}^{100}frac{sin(pi n)+1}{(-1)^n}) . sqrt{prod_{m=1}^{1000}frac{1}{cos (pi m)^{2}}}”

“= (3 + 0) times 1”“= 3 times 1”“= 3”