State whether the following statement is true or false- cos A-sin A-1 cos A-sin A-1 - cosec A- A- by using the identity cosec 2 A-1- 2 A-

The correct option is A True LHS=(cosA sinA)+1(cosA+sinA) 1 taking conjugate =(cosA sinA)+1(cosA+sinA) 1 ×(cosA+sinA)+1(cosA+sinA)+1 multiplying we get ...

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Byju's Answer

Standard XII

Mathematics

Basic Trigonometric Identities

State whether...

Question

State whether the following statement is true or false.
$\frac{cos A - sin A + 1}{cos A + sin A - 1} = cosec A + cot A$ . (by using the identity ${cosec}^{2} A = 1 + {cot}^{2} A .)$

True

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False

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Solution

The correct option is A True
$L H S = \frac{(c o s A - s i n A) + 1}{(c o s A + s i n A) - 1}$

taking conjugate

$= \frac{(c o s A - s i n A) + 1}{(c o s A + s i n A) - 1} \times \frac{(c o s A + s i n A) + 1}{(c o s A + s i n A) + 1}$

multiplying we get

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

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

on solving we get

$= \frac{2 c o s^{2} A + 2 c o s A}{2 s i n A c o s A}$

taking $2$ common and divided, we get

$= \frac{c o s A + c o s^{2} A}{s i n A c o s A}$

we can also write in the form of

$= \frac{c o s A}{s i n A c o s A} + \frac{c o s^{2} A}{s i n A c o s A}$

on solving we get

$c o s e c A + c o t A = R . H . S$

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Similar questions

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

using the identity

c o s e c^{2} A = 1 + cot^{2} A

\frac{cos A - sin A + 1}{cos A + sin A - 1} = cos e c A + cot A,

using the identity

cos e c^{2} A = 1 + {cot}^{2} A .

Q. Prove that

\frac{cos A + sin A - 1}{cos A - sin A + 1} = \frac{1}{cosec A + cot A}

, using the identity

{cosec}^{2} A - {cot}^{2} A = 1.

State whether the following statement is true or false.

\frac{sin A}{sec A + tan A - 1} + \frac{cos A}{cosec A + cot A - 1} = 1

\frac{c o s A - s i n A + 1}{c o s A + s i n - 1} = c o s e c A + c o t A,

, using the identity

c o s e c^{2} A = 1 + c o t^{2} A

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State whether the following statement is true or false.cosA−sinA+1cosA+sinA−1=cosecA+cotA. (by using the identity cosec2A=1+cot2A.)

State whether the following statement is true or false.
$\frac{cos A - sin A + 1}{cos A + sin A - 1} = cosec A + cot A$ . (by using the identity ${cosec}^{2} A = 1 + {cot}^{2} A .)$