I cosec -cot -cosec -cot - cosec - cot - 2- 1- 2 cot2 - 2 cosec - cot - ii sec -tan -sec -tan - sec -tan -2- 1-2tan 2-2 sec - tan -

(i) cosec θ+cot θ/cosec θ cot θ= (cosec θ+ cot θ) ^2= 1+ 2 cot^2 θ+ 2 cosec θ cot θ cosec θ+cot θ/cosec θ cot θ= cosec θ+cot θ/cosec θ cot θ×cosec θ+cot θ/cose ...

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Byju's Answer

Standard X

Mathematics

Trigonometric Identities

i cosec θ+cot...

Question

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

(ii) $\frac{s e c θ + t a n θ}{s e c θ - t a n θ} = (s e c θ + t a n θ)^{2} = 1 + 2 t a n^{2} θ + 2 s e c θ t a n θ$

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Solution

(i) $\frac{c o s e c θ + c o t θ}{c o s e c θ - c o t θ} = (c o s e c θ + c o t θ)^{2} = 1 + 2 c o t^{2} θ + 2 c o s e c θ c o t θ \frac{c o s e c θ + c o t θ}{c o s e c θ - c o t θ} = \frac{c o s e c θ + c o t θ}{c o s e c θ - c o t θ} \times \frac{c o s e c θ + c o t θ}{c o s e c θ + c o t θ} = \frac{(c o s e c θ + c o t θ)^{2}}{c o s e c^{2} θ - c o t^{2} θ} = \frac{(c o s e c θ + c o t θ)^{2}}{1} = (c o s e c θ + c o t θ)^{2} [F i r s t t e r m] = c o s e c^{2} θ + c o t^{2} θ + 2 c o s e c θ c o t θ = (1 + c o t^{2} θ) + c o t^{2} θ + 2 c o t θ c o s e c θ = 1 + 2 c o t^{2} θ + 2 c o t θ c o s e c θ [S e c o n d t e r m]$

(ii) $\frac{s e c θ + t a n θ}{s e c θ - t a n θ} = (s e c θ + t a n θ)^{2} = 1 + 2 t a n^{2} θ + 2 s e c θ t a n θ \frac{s e c θ + t a n θ}{s e c θ - t a n θ} = \frac{s e c θ + t a n θ}{s e c θ - t a n θ} \times \frac{s e c θ + t a n θ}{s e c θ + t a n θ} = \frac{(s e c θ + t a n θ)^{2}}{s e c^{2} θ - t a n^{2} θ} = (s e c θ + t a n θ)^{2} [F i r s t t e r m] = s e c^{2} θ + t a n^{2} θ + 2 s e c θ t a n θ = 1 + t a n^{2} θ + t a n^{2} θ + 2 s e c A t a n A = 1 + 2 t a n^{2} A + 2 s e c θ t a n θ [S e c o n d t e r m]$

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Similar questions

Q. If

\frac{sec θ - tan θ}{sec θ + tan θ} = \frac{36}{49}

, find

\frac{c o s e c θ - sec θ}{c o s e c θ + sec θ}

Q. Prove the following identity:

\frac{tan θ}{sec θ + 1} + \frac{cot θ}{c o s e c θ + 1} = c o s e c θ + sec θ - sec θ . c o s e c θ

Q. If

sec θ = \sqrt{2}

and

\frac{3 π}{2} < θ < 2 π

, then find the value of

\frac{1 + tan θ + cosec θ}{1 + cot θ - cosec θ}

(i) $s e c θ (1 - s i n θ) (s e c θ + t a n θ) = 1$
(ii) $s i n θ (1 + t a n θ) + c o s θ (1 + c o t θ) = (s e c θ + c o s e c θ)$

Q. Prove the identity

\frac{sec θ - tan θ}{sec θ + tan θ} = 1 - 2 sec θ tan θ + 2 {tan}^{2} θ

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(i) cosecθ+cotθcosecθ−cotθ=(cosecθ+cotθ)2=1+2cot2θ+2cosecθcotθ (ii) secθ+tanθsecθ−tanθ=(secθ+tanθ)2=1+2tan2θ+2secθtanθ

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

(ii) $\frac{s e c θ + t a n θ}{s e c θ - t a n θ} = (s e c θ + t a n θ)^{2} = 1 + 2 t a n^{2} θ + 2 s e c θ t a n θ$