Q.5) Prove the following identities where the angles involved
are acute angles for which the expressions are defined.
EXERCISE 8.4
Proof:
(x)
1
+
tan2A
1
+
cot2A
=
1
–
tan A
1
–
cot A
2
=
tan2A
1
+
tan2A
1
+
cot2A
=
cosec2A
sec2A
=
1
cos2A
×
sin2A
1
1
cos2A
1
sin2A
=
cos2A
sin2A
=
tan2A
=
1
+
tan2A
1
+
cot2A
=
∴
tan2A
…(i)
1 + tan2θ = sec2θ
1 + cot2θ = cosec2θ
÷
1
–
tan A
1
–
cot A
=
tan²A
…(ii)
2
From (i) and (ii),
∴
1
+
tan2A
1
+
cot2A
=
1
–
tan A
1
–
cot A
2
=
tan2A
∴
Q.5) Prove the following identities where the angles involved
are acute angles for which the expressions are defined.
EXERCISE 8.4
Proof:
(x)
1
+
tan2A
1
+
cot2A
=
1
–
tan A
1
–
cot A
2
=
tan2 A
=
cos A
–
sin A
cos A
÷
sin A
–
cos A
2
sin A
2
1
–
tan A
1
–
cot A
2
=
1
–
sin A
cos A
÷
1
–
cos A
sin A
2
2
=
(sin A
–
cos A)2
cos2A
×
sin2A
(sin A
–
cos A)2
=
sin2A
cos2A
1
+
tan2A
1
+
cot2A
=
tan2A
…(i)
=
(sin A
–
cos A)
cos A
÷
sin A
–
cos A
2
sin A
2
–
a – b = – (b – a)