Question

(18) Prove that cosA-sinA+1 / cosA+sinA-1 = cosecA + cotA = 1+cosA / sinA


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Answer 1

Answer:

ANSWER

LHS=

cosA+sinA−1

cosA−sinA+1

dividing Nr and Dr by sinA we get,

=

sinA

cosA

+

sinA

sinA

−

sinA

1

sinA

cosA

−

sinA

sinA

+

sinA

1

=

cotA+1−cosecA

cotA−1+cosecA

=

cotA+1−cosecA

cotA+cosecA−(cosec

2

A−cot

2

A)

=

cotA+1−cosecA

(cotA+cosecA)(1−cosecA+cotA)

=cotA+cosecA=RHS

Step-by-step explanation:

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Answer 2

Given :

Prove that :

$\\ \pink \bigstar \rm \: \frac{cosx - sinx + 1}{cosx + sinx - 1} = cosecx + cotx$

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Formulas :

$\pink \bigstar \bf \: {cosec }^{2} \theta - {cot}^{2} \theta = 1$

$\\ \red \bigstar \bf {sec}^{2} \theta - {tan}^{2} \theta = 1$

$\blue \bigstar \bf \: {sin}^{2} \theta + {cos}^{2} \theta = 1$

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Solution :

From LHS:

$\\ \longrightarrow \sf \frac{cosx - sinx + 1}{cosx + sinx - 1} \\ \\ \therefore \bf \: we \: know \: that \: \\ \pink \bigstar \bf \: {cosec}^{2} \theta - {cot}^{2} \theta = 1 \\ \pink \bigstar \bf \: {sec}^{2} \theta - {tan }^{2} \theta = 1 \\ \pink \bigstar \bf{sin}^{2} \theta + {cos}^{2} \theta = 1$

Hence,

$\\ \longrightarrow \sf \: \frac{cosx - sinx + 1}{cosx + sinx - 1} \\ \\ \\ \longrightarrow \sf \: \frac{ \lgroup \: cosx + (1 - sinx) \rgroup}{ \lgroup \: cosx - (1 - sinx) \rgroup} \times \frac{ \lgroup \: cosx + (1 - sinx) \rgroup}{ \lgroup \: cosx + (1 - sinx) \rgroup} \\ \\ \\ \longrightarrow \sf \: \frac{ { \lgroup \: cosx + (1 - sinx) \rgroup}^{2} }{ \lgroup {cos}^{2} x - (1 - {sin}^{2} x) \rgroup} \\ \\ \\ \longrightarrow \sf \: \frac{ {cos}^{2} x + 1 + {sin}^{2} x - 2sinx + 2cosx(1 - sinx)}{ {cos}^{2} x - 1 - {sin}^{2} x + 2sinx} \\ \\ \\ \longrightarrow \sf \: \frac{2 - 2sinx + 2cosx(1 - sinx)}{ - 2 {sin}^{2} x + 2sinx} \\ \\ \\ \longrightarrow \sf \: \frac{2(1 - sinx)(1 + cosx)}{2sinx(1 - sinx)} \\ \\ \\ \longrightarrow \sf \: \frac{(1 + cosx)}{sinx} \\ \\ \longrightarrow \sf \: \frac{1}{sinx} + \frac{cosx}{sinx} \\ \\ \longrightarrow \boxed{ \sf{ cosecx + cotx}}$

RhS:

$\green \bigstar \sf \: cosecx + cotx$

We can see that LHS=RHS, so we can write.

$\\ \blue \bigstar \boxed{ \sf{ \frac{cosx - sinx + 1}{cosx + sinx - 1} = cosecx + cotx}}$

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Properties :

$\bf \: (1) \: \frac{1}{sinx} = cosecx \\ \\ \bf \:( 2) \: \frac{1}{cosecx} = sinx \\ \\ \bf \: (3) \: tanx = \frac{sinx}{cosx} = \frac{1}{cotx} \\ \\ \bf (4)\: cosx = \frac{1}{secx}$