Evaluate 1-sinx-1-sinx

Consider 1+sinx/1 sinxOn multiply and divide by 1+sinx i.e,1+sinx/1 sinx×1+sinx/1+sinx0ex0ex(1+sinx)^2/1^2 sin^2x0ex0ex(1+sinx)^2/cos^2x0ex0ex((1+sinx)/cosx)^20 ...

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Byju's Answer

Standard XII

Mathematics

Parametric Differentiation

Evaluate 1+si...

Question

Evaluate $\frac{1 + \sin x}{1 - \sin x}$

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Solution

Consider $\frac{1 + \sin x}{1 - \sin x}$
On multiply and divide by $1 + \sin x$ i.e,
$\frac{1 + \sin x}{1 - \sin x} \times \frac{1 + \sin x}{1 + \sin x} \frac{{(1 + \sin x)}^{2}}{1^{2} - \sin^{2} x} \frac{{(1 + \sin x)}^{2}}{\cos^{2} x} {(\frac{(1 + \sin x)}{\cos x})}^{2} {(\frac{1}{\cos x} + \frac{\sin x}{\cos x})}^{2} {(s e c x + \tan x)}^{2}$
Hence, $\frac{1 + \sin x}{1 - \sin x} = {(s e c x + \tan x)}^{2}$

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Evaluate the expression when $x = - \frac{4}{5} a n d y = \frac{1}{3}$

$2 x + 6 y$

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. 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.

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

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Evaluate 1+sinx1-sinx

Consider 1+sinx1-sinxOn multiply and divide by 1+sinx i.e,1+sinx1-sinx×1+sinx1+sinx1+sinx212-sin2x1+sinx2cos2x1+sinxcosx21cosx+sinxcosx2secx+tanx2Hence, 1+sinx1-sinx=secx+tanx2

Evaluate $\frac{1 + \sin x}{1 - \sin x}$