9 ⋅5 Trigonometric identity
(i)(sinθ)^2 (cosθ)^2 =2 2
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OPOM
OPPM= 2
2
22
OPOM
OPPM
= 22 2
OPPM OM
= 22
OPOP
[ by the formula of Pythagoras ]1
or,(sinθ)^2 (cosθ)^2 1
? sin^2 θcos^2 θ 1Remark : For integer indices n we can write sinnθ for (sinθ)n and cosnθ for (cosθ)n.
(ii) sec^2 θ=(secθ)^2 =2
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OMOP= 22OMOP
= 22 2OMPM OM
[OP is the hypotenuse of right angled 'POM]= 22
22
OMOM
OMPM
=
2(^1) ̧
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OM
PM
= 1 (tanθ)^2 = 1 tan^2 θ
? sec^2 θ 1 tan^2 θ
or, sec^2 θtan^2 θ 1
or, tan^2 θ sec^2 θ 1
(iii)cosec^2 θ (cosecθ)^2 =
2
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PM
OP
= 2
2
PM
OP
= 2
2 2
PM
PM OM
[i is the hypotenuse of right-angled 'POM]
= 2
2
2
2
PM
OM
PM
PM
=
2
(^1) ̧
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§
PM
OM
= 1 (cotθ)^2 = 1 cot^2 θ
? cosec^2 θcot^2 θ 1 and cot^2 θ cosec^2 θ 1