sin theta + cos theta is equal to cosec theta prove LHS and RHS equal
Answers
Answer:
We have to prove that
cot+cosecθ
sinθ
=2+
cotθ−cosecθ
sinθ
or,
cotθ+cosecθ
sinθ
−
cotθ−cosecθ
sinθ
=2
Now,
⇒ LHS =
cotθ+cosecθ
sinθ
−
cotθ−cosecθ
sinθ
⇒ LHS =
cosecθ+cotθ
sinθ
+
cosecθ−cotθ
sinθ
⇒ LHS = sinθ{
cosecθ+cotθ
1
+
cosecθ−cotθ
1
}
⇒ LHS = $$sin \, \theta\left\{\dfrac{cosec \, \theta \, - \, cot \, \theta \, + \, cosec \, \theta \, + \, cot \, \theta}{cosec^2 \, \theta \, - \, cot^ \, \theta}\right\} \, = \, sin \, \theta \, \left(\dfrac{2 \, cosec \, \theta}{1} \right)$$
⇒ LHS = sinθ(2cosecθ)=2sinθ×
sinθ
1
=2=RHS
⇒ LHS = 2 = RHS
ALTERNANATIVELY,
LHS =
cotθ+cosecθ
sinθ
⇒LHS=sinθ(cosecθ−cotθ) [∵
cosecθ+cotθ
1
=cosecθ−cotθ]
⇒ LHS = sinθ(
sinθ
1
−
sinθ
cosθ
)=sinθ(
sinθ
1−cosθ
)
⇒ LHS = 1 - cos θ
⇒ = 2 - (1 + cos θ)
⇒ LHS = 2 -
1−cosθ
(1+cosθ)(1−cosθ)
⇒ LHS = 2 -
1−cosθ
(1−cos
2
θ)
⇒ LHS = 2 -
1−cosθ
sin
2
θ
=2−
sinθ
1−cosθ
sinθ
=2−
sinθ
1
−
sinθ
cosθ
sinθ
⇒ LHS = 2 -
cosecθ−cotθ
sinθ
= RHS
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