14. Prove that: Sin A (1 + tan A) + cos A (1 + cot A) = Sec A + Cosec A
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sin∅( 1 + tan ∅ ) + cos∅( 1 + cot ∅ ) = ( sec∅ + cosex∅ ) .
▶ Step-by-step explanation :-
Solving LHS :-)
∵sinθ(1+tanθ)+cosθ(1+cotθ)
=sinθ+sinθ×
cosθ
sinθ
+cosθ+cosθ×
sinθ
cosθ
.
=
sinθcosθ
sin
2
θcosθ+sin
2
θ+cos
2
θsinθ+cos
2
θ
=
cosθsinθ
(sin
3
θ+cos
3
θ)+(cosθsin
2
θ+cos
2
θsinθ)
.
=
cosθsinθ
(sinθ+cosθ)(sin
2
θ−sinθcosθ+cos
2
θ)+sinθcosθ(sinθ+cosθ)
.
=
cosθsinθ
(sinθ+cosθ)(sin
2
θ+cos
2
θ
−sinθcosθ
+sinθcosθ
)
.
=
cosθsinθ
(sinθ+cosθ)×1.
.
=
cosθ
sinθ
sinθ
+
cosθ
sinθ
cosθ
.
=
cosθ
1
+
sinθ
1
.
=secθ+cosecθ.
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