Abbreviations used :
L.H.S -----> Left hand side
R.H.S -----> Right hand side
Example 1 :
Prove :
tanθ/(1 - tan2θ) = sinθsin(90 - θ)/[2sin2(90 - θ) - 1]
Solution :
L.H.S
= tan θ/(1 - tan2θ)
= sinθsin(90 - θ)/[2sin2(90 - θ) - 1]
= R.H.S
Example 2 :
Prove :
[1/(cosecθ - cotθ)] - (1/sinθ)
= [(1/sinθ)] - [1/(cosecθ + cotθ)]
Solution :
L.H.S :
= [1/(cosecθ - cotθ)] - (1/sinθ)
Multiply numerator and denominator of the first fraction by (cosecθ + cotθ).
= [(1/(cosecθ - cotθ)) x (cosecθ + cotθ)/(cosecθ + cotθ)] - (1/sinθ)
= (cosecθ + cotθ)/(cosec2θ - cot2θ) - (1/sinθ)
= [(cosecθ + cotθ)/1] - (1/sinθ)
= [(cosecθ + cotθ) - cosecθ]
= cotθ -----(1)
R.H.S :
= (1/sinθ) - [1/(cosecθ - cotθ)]
Multiply the numerator and denominator of the second fraction by (cosecθ + cotθ).
= [(cosecθ - cotθ)/(cosecθ - cotθ)(cosecθ + cotθ)] - (1/sinθ)
= (cosecθ + cotθ)/(cosec2θ - cot2θ) - (1/sinθ)
= [(cosecθ + cotθ)/1] - (1/sinθ)
= [(cosecθ + cotθ) - cosecθ]
= cotθ -----(2)
From (1) and (2), we get
L.H.S = R.H.S
Example 3 :
Prove :
(cot2θ + sec2θ) / (tan2θ + cosec2θ)
= sinθcosθ(tanθ + cotθ)
Solution :
L.H.S :
= (cot2θ + sec2θ)/(tan2θ + cosec2θ)
= (cosec2θ - 1 + 1 + tan2θ)/(tan2θ + cosec2θ)
= (tan2θ + cosec2θ)/(tan2θ + cosec2θ)
= 1 -----(1)
R.H.S :
= sinθcosθ(tanθ + cotθ)
= sinθcosθ[(sinθ/cosθ) + (cosθ/sinθ)]
= sinθcosθ[(sin2θ + cos2θ)/(cosθsinθ)]
= sinθcosθ [1/(cosθsinθ)]
= 1 -----(2)
From (1) and (2), we get
L.H.S = R.H.S
Example 4 :
If x = asecθ + btanθ and y = atanθ + bsecθ then prove that
x2 - y2 = a2 - b2
Solution :
x2 = (asecθ + btanθ)2
x2 = (asecθ)2 + (btanθ)2 + 2absecθtanθ
x2 = a2sec2θ + b2tan2θ + 2absecθtanθ -----(1)
y2 = (atanθ + bsecθ)2
y2 = (atanθ)2 + (bsecθ)2 + 2abtanθsecθ
y2 = a2tan2θ + b2sec2θ + 2abtanθsecθ -----(2)
(1) - (2) :
x2 - y2 = (a2sec2θ + b2tan2θ + 2absecθtanθ)
- (a2tan2θ + b2sec2θ + 2abtanθsecθ)
x2 - y2 = a2sec2θ + b2tan2θ + 2absecθtanθ
- a2tan2θ - b2sec2θ - 2abtanθsecθ
x2 - y2 = a2sec2θ + b2tan2θ - a2tan2θ - b2sec2θ
x2 - y2 = a2sec2θ - a2tan2θ - b2sec2θ + b2tan2θ
x2 - y2 = a2(sec2θ - tan2θ) - b2(sec2θ - tan2θ)
x2 - y2 = a2(1) - b2(1)
x2 - y2 = a2 - b2
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