1 of 3

L.H.S

=

=

×

Q.

cosec x

–1

cosec x

+1

=

1

sec x

+

tan x

 

 

 

(cosec x

1)

–

(cosec x

1)

+

(cosec x

1)

–

(cosec x

1)

+

(cosec x

1)

–

(cosec x

1)

–

 

(cosec x

1)2

–

cosec2x

1

–

=

 

(cosec x

1)2

–

cot2x

=

cosec x

1

–

cot x

=

L.H.S

Proof.

cosec2θ = 1 + cot2θ

∴ cosec2θ – 1 = cot2θ

(a – b) (a + b) = a2 – b2

2 of 3

L.H.S

=

Q.

cosec x

–1

cosec x

+1

=

1

sec x

+

tan x

 

Proof.

cosec x

cot x

–1

=

cosec x

cot x

–

1

cot x

=

1

sin x

÷

cos x

sin x

–

1

cot x

=

1

sin x

×

sin x

cos x

–

1

cot x

=

1

cos x

–

tan x

=

sec x

–

tan x

=

(cosec x

÷

cot x)

–

1

cot x

 

 

 

 

3 of 3

L.H.S

=

Q.

cosec x

–1

cosec x

+1

=

1

sec x

+

tan x

 

Proof.

sec x

tan x

–

=

(sec x

tan x)

–

(sec x

tan x)

+

(sec x

tan x)

+

=

sec 2x

tan2x

–

(sec x

tan x)

+

=

1

(sec x

tan x)

+

=

R.H.S

 

(cosec x

1)

–

(cosec x

1)

+

=

1

(sec x

tan x)

+

L.H.S

L.H.S

∴

∴

(a – b) (a + b) = a2 – b2

sec2θ – tan2θ = 1