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The simplified value of (secA - cosA)2 + (cosecA - sinA)2 - (cotA - tanA)2
Answer & Solution
Correct Answer:
Option
C
(secA - cosA)2 + (cosecA - sinA)2 - (cotA - tanA)2
= (sec2A + cos2A - 2secA.cosA) + (coses2A + sin2A - 2cosecA.sinA) - (cot2A + tan2A - 2cotA.tanA)
= sec2A - tan2A + cos2A + sin2A + coses2A - cot2A - 2
= 3 - 2
= 1
Alternate shortcut method:
(secA - cosA)2 + (cosecA - sinA)2 - (cotA - tanA)2
Put θ = 45°
= (sec45° - cos45°)2 + (cosec45° - sin45°)2 - (cot45° - tan45°)2
$$\eqalign{ & = {\left( {\sqrt 2 - \frac{1}{{\sqrt 2 }}} \right)^2} + {\left( {\sqrt 2 - \frac{1}{{\sqrt 2 }}} \right)^2} - {\left( {1 - 1} \right)^2} \cr & = \frac{1}{2} + \frac{1}{2} - 0 \cr & = 1 \cr} $$
= (sec2A + cos2A - 2secA.cosA) + (coses2A + sin2A - 2cosecA.sinA) - (cot2A + tan2A - 2cotA.tanA)
= sec2A - tan2A + cos2A + sin2A + coses2A - cot2A - 2
= 3 - 2
= 1
Alternate shortcut method:
(secA - cosA)2 + (cosecA - sinA)2 - (cotA - tanA)2
Put θ = 45°
= (sec45° - cos45°)2 + (cosec45° - sin45°)2 - (cot45° - tan45°)2
$$\eqalign{ & = {\left( {\sqrt 2 - \frac{1}{{\sqrt 2 }}} \right)^2} + {\left( {\sqrt 2 - \frac{1}{{\sqrt 2 }}} \right)^2} - {\left( {1 - 1} \right)^2} \cr & = \frac{1}{2} + \frac{1}{2} - 0 \cr & = 1 \cr} $$
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