Prove that -1 - tan - - sec - 1 - cot - - cosec - - 2

(1 + tan θ nbsp; + sec θ ) (1 + cot θ nbsp; cosec θ ) = (1 + sin θ/cos θ + 1/cos θ)(1 + cos θ/sin θ 1/sin θ) = ((cos θ + sin θ + 1)/cos θ)((cos θ + si ...

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Byju's Answer

Standard XII

Mathematics

General Solution of Trigonometric Equation

Prove that :1...

Question

Prove that :
$(1 + t a n θ + s e c θ) (1 + c o t θ - c o s e c θ)$ = 2

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Solution

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

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

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

$\Rightarrow 2$
Hence proved

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Similar questions

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

(sec θ - cos θ)^{2} + (c o s e c θ - sin θ)^{2} - (cot θ - tan θ)^{2} =

[1 mark]

Q. Question 4 (ii)
Choose the correct option.
(ii)

(1 + t a n θ + s e c θ) (1 + c o t θ - c o s e c θ)

(A) 0
(B) 1
(C) 2
(D) -1

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Prove that : (1+tanθ+secθ)(1+cotθ−cosecθ) = 2

Prove that :
$(1 + t a n θ + s e c θ) (1 + c o t θ - c o s e c θ)$ = 2