Advertisements
Advertisements
Question
If θ1, θ2, θ3, ..., θn are in A.P., whose common difference is d, show that secθ1 secθ2 + secθ2 secθ3 + ... + secθn–1 . secθn = `(tan theta_n - tan theta_1)/sin d`
Solution
Since θ1, θ2, θ3, ..., θn are in A.P.
∴ θ2 – θ1 = θ3 – θ2 = ... = θn – θn-1 = d
Now we have to prove that
secθ1 secθ2 + secθ2 secθ3 + ... + secθn–1 . secθn = `(tan theta_n - tan theta_1)/sind` L.H.S.
⇒ `sin d/sin d [sec theta_1 * sec theta_2 + sec theta_2 sec theta_3 + ... + sec theta_(n - 1) * sec theta_n]`
Taking only `(sind[sec theta_1 * sec theta_2])/sind = (sind[1/cos theta_1 * 1/cos theta_2])/sind`
= `(sin(theta_2 - theta_1))/sind * 1/(costheta_1 costheta_2)`
= `1/sind [(sin theta_2 cos theta_1 - cos theta_2 sin theta_1)/(cos theta_1 cos theta_2)]`
= `1/sind [(sin theta_2 cos theta_1)/(cos theta_1 cos theta_2) - (cos theta_2 sin theta_1)/(cos theta_1 cos theta_2)]`
= `1/sind [tan theta_2 - tan theta_1]`
Similarly we can solve other terms which will be
`1/sind [tan theta_3 - tan theta_2]`
And `1/sind [tan theta_4 - tan theta_3]`
Here L.H.S. = `1/sind [tan theta_2 - tan theta_1 + tan theta_3 - tan theta_2 + ... + tan theta_n - tan theta_(n - 1)]`
= `1/sind [- tan theta_1 + tan theta_n]`
= `(tan theta_n - tan theta_1)/sind` R.H.S.
L.H.S. = R.H.S.
Hence proved.
APPEARS IN
RELATED QUESTIONS
Write the first five terms of the sequences whose nth term is:
`a_n = n(n+2)`
Write the first five terms of the sequences whose nth term is:
`a_n = n/(n + 1)`
Write the first five terms of the sequences whose nth term is:
an = 2n
Write the first five terms of the sequences whose nth term is:
`a_n = (2n -3)/6`
Write the first five terms of the sequences whose nth term is:
`a_n = (-1)^(n-1) 5^(n+1)`
Write the first five terms of the sequences whose nth term is:
`a_n = n (n^2 + 5)/4`
Find the indicated term in the following sequence whose nth term is:
an = 4n – 3; a17, a24
Find the indicated term in the following sequence whose nth term is:
`a_n = n^2/2^n`; `a_7`
Find the indicated term in the following sequence whose nth term is:
`a_n = (–1)^(n – 1) n^3; a_9`
Find the indicated term in the following sequence whose nth term is:
`a_n = (n(n-2))/(n+3)` ;`a_20`
If S1, S2, S3 are the sum of first n natural numbers, their squares and their cubes, respectively, show that `9S_2^2 = S_3(1 + 8S_1)`
A sequence may be defined as a ______.
Two sequences cannot be in both A.P. and G.P. together.
Every progression is a sequence but the converse, i.e., every sequence is also a progression need not necessarily be true.
| Column I | Column II |
| (a) `4, 1, 1/4, 1/16` | (i) A.P |
| (b) 2, 3, 5, 7 | (ii) Sequence |
| (c) 13, 8, 3, –2, –7 | (iii) G.P. |
