Abbreviations used :
L.H.S -----> Left hand side
R.H.S -----> Right hand side
Example 1 :
Prove :
tanθ/(1 - cotθ) + cotθ/(1 - tanθ) = 1 + secθ cosecθ
Solution :
L.H.S :
= tanθ/(1 - cotθ) + cotθ/(1 - tanθ)
a3 - b3 = (a - b)(a2 + ab + b2)
= 1 + cosecθsecθ
= R.H.S
Example 2 :
Prove :
sin(90 - θ)/(1 - tanθ) + cos(90 - θ)/(1 - cotθ)
= cosθ + sin θ
Solution :
L.H.S :
= sin(90 - θ)/(1 - tanθ) + cos(90 - θ)/(1 - cotθ)
= cosθ/(1 - tanθ) + sinθ/(1 - cotθ)
a2 - b2 = (a + b)(a - b)
= (cosθ + sinθ)(cosθ - sinθ)/(cosθ - sinθ)
= (cosθ + sinθ)
= R.H.S
Example 3 :
Prove :
[tan(90 - θ)/(cosecθ + 1)] + [(cosecθ + 1)/cotθ)]
= 2 sec θ
Solution :
L.H.S :
= [tan(90 - θ)/(cosecθ + 1)] + [(cosecθ + 1)/cotθ)]
we can write tan(90 - θ) as cotθ.
= (2/sinθ) / [cosθ/sinθ]
= (2/sinθ) x (sinθ/cosθ)
= 2/cosθ
= 2secθ
= R.H.S
Example 4 :
Prove :
(cotθ + cosecθ - 1)/(cotθ - cosecθ + 1)
= cosecθ + cotθ
Solution :
L.H.S :
= (cotθ + cosecθ - 1)/(cotθ - cosecθ + 1)
= R.H.S
Example 5 :
Prove :
(1 + cotθ - cosecθ)(1 + tanθ + secθ) = 2
Solution :
L.H.S :
= 2sinθcosθ/sinθcosθ
= 2
= R.H.S
Example 6 :
Prove :
(sinθ - cosθ + 1)/(sinθ + cosθ - 1) = 1/(secθ - tanθ)
Solution :
L.H.S :
= (sinθ - cosθ + 1)/(sinθ + cosθ - 1)
Dividing everything by cos θ.
= R.H.S
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