without using trigonometric identities prove that ( 1+ tan A + cot A) ( sin A - cos A = sin A - cot A × cos A
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Answer:
L.H.S= (1+cotA+tanA)(sinA−cosA)
=(1+
sinA
cosA
+
cosA
sinA
)(sinA−cosA)
=(
sinAcosA
sinAcosA+cos
2
A+sin
2
A
)(sinA−cosA)
=(1+sinAcosA)(
sinAcosA
sinA−cosA
)
Now,
R.H.S=sinAtanA−cotAcosA
=sinA
cosA
sinA
−
sinA
cosA
cosA
=
cosA
sin
2
A
−
sinA
cos
2
A
=
sinAcosA
sin
3
A−cos
3
A
=
sinAcosA
(sinA−cosA)
(sin
2
A+cos
2
A+sinAcosA)
=(1+sinAcosA)
sinAcosA
(sinA−cosA)
Hence,
L.H.S=R.H.S
proved
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