1 of 2

Q.5) Prove the following identities where the angles involved

are acute angles for which the expressions are defined.

EXERCISE 8.4

(ix)

(cosec A

–

sin A)

(sec A

–

cos A)

=

1

tan A

+

cot A

Proof:

L.H.S

=

(cosec A

–

sin A)

–

cos A)

=

1

sin A

–

sin A

1

cos A

–

cos A

=

1

sin A

–

sin2A

1

cos A

–

cos2A

=

sin A

×

cos A

=

cos A .

sin A

(sec A

cos2A

sin2A

1

sin θ

cosec θ =

1

cos θ

=

sec θ

1 – sin2θ = cos2θ

1 – cos2θ = sin2θ

L.H.S

∴

…(i)

2 of 2

Q.5) Prove the following identities where the angles involved

are acute angles for which the expressions are defined.

EXERCISE 8.4

(ix)

(cosec A

–

sin A)

(sec A

–

cos A)

=

1

tan A

+

cot A

Proof:

R.H.S

=

1

tan A

+

cot A

=

1

sin A

cos A

+

cos A

sin A

=

1

sin2A

+

cos2A

cos A

×

sin A

=

cos A .

sin A

∴

L.H.S = R.H.S

=

1

1

cos A

×

sin A

L.H.S = cos A . sin A

∴

(cosec A

–

sin A)

(sec A

–

cos A)

=

1

tan A

+

cot A

 

 

sin2θ + cos2θ = 1

÷

÷

÷

R.H.S

∴

…(ii)

…[From (i) and (ii)]

…(i)