Evaluate 1-cos-1-cos-

Consider1+cos⌽/1 cos⌽Multiply and divided by 1+cos⌽1+cos⌽/1 cos⌽×1+cos⌽/1+cos⌽0ex0ex(1+cos⌽)^2/1^2 (cos⌽)^20ex0ex(1+cos⌽)^2/1 cos^2⌽0ex0ex(1+cos⌽)^2/sin^2⌽^20ex ...

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Byju's Answer

Standard XII

Mathematics

Determinant

Evaluate 1+co...

Question

Evaluate $\frac{1 + \cos ⌽}{1 - \cos ⌽}$

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Solution

Consider
$\frac{1 + \cos ⌽}{1 - \cos ⌽}$
Multiply and divided by $1 + \cos ⌽$
$\frac{1 + \cos ⌽}{1 - \cos ⌽} \times \frac{1 + \cos ⌽}{1 + \cos ⌽} \frac{{(1 + \cos ⌽)}^{2}}{1^{2} - {(\cos ⌽)}^{2}} \frac{{(1 + \cos ⌽)}^{2}}{1 - \cos^{2} ⌽} \frac{{(1 + \cos ⌽)}^{2}}{\sin^{2} ⌽^{2}} {(\frac{(1 + \cos ⌽)}{\sin ⌽})}^{2} {(\frac{1}{\sin ⌽} + \frac{\cos ⌽}{\sin ⌽})}^{2} {(\cos e c ⌽ + c o t ⌽)}^{2}$
Hence, $\frac{1 + \cos ⌽}{1 - \cos ⌽}$ is ${(\cos e c ⌽ + co t ⌽)}^{2}$

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Q. Assertion :

Δ = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \frac{(1 + a_{1} b_{1}) (1 + a_{1}^{2} b_{1}^{2} - a_{1} b_{1})}{1 + a_{1} b_{1}} & \frac{(1 + a_{1} b_{2}) (1 + a_{1}^{2} b_{2}^{2} - a_{1} b_{2})}{1 + a_{1} b_{2}} & \frac{(1 + a_{1} b_{3}) (1 + a_{1}^{2} b_{3}^{2} - a_{1} b_{3})}{1 + a_{1} b_{3}} \frac{(1 + a_{2} b_{1}) (1 + a_{2}^{2} b_{1}^{2} - a_{2} b_{1})}{1 + a_{2} b_{1}} & \frac{(1 + a_{2} b_{2}) (1 + a_{2}^{2} b_{2}^{2} - a_{2} b_{2})}{1 + a_{2} b_{2}} & \frac{(1 + a_{2} b_{3}) (1 + a_{2}^{2} b_{3}^{2} - a_{2} b_{3})}{1 + a_{2} b_{3}} \frac{(1 + a_{3} b_{1}) (1 + a_{3}^{2} b_{1}^{2} - a_{3} b_{1})}{1 + a_{3} b_{1}} & \frac{(1 + a_{3} b_{2}) (1 + a_{2}^{2} b_{2}^{2} - a_{3} b_{2})}{1 + a_{3} b_{2}} & \frac{(1 + a_{3} b_{3}) (1 + a_{3}^{2} b_{3}^{2} - a_{3} b_{3})}{1 + a_{3} b_{3}} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

Δ = 0

Reason:

Δ

can be written as product of two determinants.

Q. If

ω

is complex cube root of unity and a, b, c are three real numbers such that

\frac{1}{a + ω} + \frac{1}{b + ω} + \frac{1}{c + ω} = 2 ω^{2}

and

\frac{1}{a + ω^{2}} + \frac{1}{b + ω^{2}} + \frac{1}{c + ω^{2}} = 2 ω

then

\frac{1}{a + 1} + \frac{1}{b + 1} + \frac{1}{c + 1}

$l o g_{e} (n + 1) - l o g_{e} (n - 1) = 4 a [(\frac{1}{n}) + (\frac{1}{3 n^{3}}) + (\frac{1}{5 n^{5}}) + . . . \infty]$ Find $8 a$ .

Q. If

α

β

γ

are the roots of

x^{3} + a x^{2} + b = 0

b \neq 0

then the determinant

Δ

, where

Δ = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} \frac{1}{α} & \frac{1}{β} & \frac{1}{γ} \frac{1}{β} & \frac{1}{γ} & \frac{1}{α} \frac{1}{γ} & \frac{1}{α} & \frac{1}{β} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣

equals

Q. Consider the sequence

a_{n}

given by

a_{1} = \frac{1}{2}, a_{n} + 1 = a_{n}^{2} + a_{n}

Let

S_{n} = \frac{1}{a_{1} + 1} + \frac{1}{a_{2} + 1} + . . . . . . . . . . . + \frac{1}{a_{n} + 1}

then find the value of

[S_{2012}]

, where [.] denotes greatest integer function.

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Evaluate 1+cos⌽1-cos⌽

Consider1+cos⌽1-cos⌽Multiply and divided by 1+cos⌽1+cos⌽1-cos⌽×1+cos⌽1+cos⌽1+cos⌽212-cos⌽21+cos⌽21-cos2⌽1+cos⌽2sin2⌽21+cos⌽sin⌽21sin⌽+cos⌽sin⌽2cosec⌽+cot⌽2Hence, 1+cos⌽1-cos⌽ is cosec⌽+cot⌽2

Evaluate $\frac{1 + \cos ⌽}{1 - \cos ⌽}$