Cosec- - cot-2 - - a 1-cos-1- cos- b 1-cos-1- cos- c 1-sin-1- sin- d none of these

(cosec θ cot θ)^2 =cot^2θ 2cotθcosecθ+cosec^2θ = cos^2θ/sin^2θ 2 (cosθ/sinθ)(1/sinθ)+ 1/sin^2θ =cos^2θ/sin^2θ 2cosθ/sin^2θ+1/sin^2θ =(cos^2θ 2cosθ+1)/sin^2θ ...

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Standard X

Mathematics

Trigonometric Identities

cosec θ - cot...

Question

$(c o s e c θ - c o t θ)^{2}$ = ?

(a) $\frac{1 + c o s θ}{1 - c o s θ}$ (b) $\frac{1 - c o s θ}{1 + c o s θ}$ (c) $\frac{1 + s i n θ}{1 - s i n θ}$ (d) none of these

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Solution

$(c o s e c θ - c o t θ)^{2}$

$= c o t^{2} θ - 2 c o t θ c o s e c θ + c o s e c^{2} θ$

$= \frac{c o s^{2} θ}{s i n^{2} θ} - 2 (\frac{c o s θ}{s i n θ}) (\frac{1}{s i n θ}) + \frac{1}{s i n^{2} θ}$

$= \frac{c o s^{2} θ}{s i n^{2} θ} - \frac{2 c o s θ}{s i n^{2} θ} + \frac{1}{s i n^{2} θ}$

$= \frac{(c o s^{2} θ - 2 c o s θ + 1)}{s i n^{2} θ}$

$= \frac{(1 - c o s θ)^{2}}{(1 - c o s^{2} θ)}$

$= \frac{(1 - c o s θ)^{2}}{(1 + c o s θ) (1 - c o s θ)}$

$= \frac{(1 - c o s θ)}{(1 + c o s θ)}$

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Similar questions

Prove the following trigonometric identities.
(i) $\frac{1 + cos θ + sin θ}{1 + cos θ - sin θ} = \frac{1 + sin θ}{cos θ}$

(ii) $\frac{sin θ - cos θ + 1}{sin θ + cos θ - 1} = \frac{1}{sec θ - tan θ}$

(iii) $\frac{cos θ - sin θ + 1}{cos θ + sin θ - 1} = cosec θ + cot θ$

(iv) $(sin θ + cos θ) (tan θ + cot θ) = sec θ + cosec θ$

Q. Prove that

\frac{cos θ - sin θ + 1}{cos θ + sin θ - 1} = c o s e c θ + cot θ

Q. The numerical value of

cosec θ [\frac{1 - cos θ}{sin θ} + \frac{sin θ}{1 - cos θ}] - 2 {cot}^{2} θ

The expression $E = \frac{1 + cos θ + sin θ}{1 + cos θ - sin θ}$ can be simplified to

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(cosec θ−cot θ)2 = ? (a) 1+cos θ1−cos θ (b) 1−cos θ1+cos θ (c) 1+sin θ1−sin θ (d) none of these

(cosec θ−cot θ)2 =cot2θ−2cotθcosecθ+cosec2θ =cos2θsin2θ−2(cosθsinθ)(1sinθ)+1sin2θ =cos2θsin2θ−2cosθsin2θ+1sin2θ =(cos2θ−2cosθ+1)sin2θ =(1−cosθ)2(1−cos2θ) =(1−cosθ)2(1+cosθ)(1−cosθ) =(1−cosθ)(1+cosθ)

$(c o s e c θ - c o t θ)^{2}$ = ?

(a) $\frac{1 + c o s θ}{1 - c o s θ}$ (b) $\frac{1 - c o s θ}{1 + c o s θ}$ (c) $\frac{1 + s i n θ}{1 - s i n θ}$ (d) none of these