if cos theta + sin theta is equal to root 2 cos theta then prove that cos theta minus sin theta equal to root 2 Sin Theta
Answers
Answered by
1
Answer:
sin
θ
+
cos
θ
=
−
√
2
cos
θ
Explanation:
Here,
sin
θ
−
cos
θ
=
√
2
sin
θ
⇒
sin
θ
−
√
2
sin
θ
=
cos
θ
⇒
sin
θ
(
1
−
√
2
)
=
cos
θ
⇒
sin
θ
[
(
1
−
√
2
)
(
1
+
√
2
1
+
√
2
)
]
=
cos
θ
⇒
sin
θ
[
1
−
2
1
+
√
2
)
=
cos
θ
⇒
sin
θ
(
−
1
)
=
(
1
+
√
2
)
cos
θ
⇒
−
sin
θ
=
cos
θ
+
√
2
cos
θ
⇒
−
sin
θ
−
cos
θ
=
√
2
cos
θ
⇒
sin
θ
+
cos
θ
=
−
√
2
cos
θ
Answered by
3
Step-by-step explanation:
We have,
→ cos θ + sin θ = √2cos θ .
[ Squaring both side, we get ] .
⇒ ( cos θ + sin θ )² = 2cos²θ .
⇒ cos²θ + sin²θ + 2cosθsinθ = 2cos² .
⇒ sin²θ + 2cosθsinθ = 2cos²θ - cos²θ .
⇒ sin²θ + 2cosθsinθ = cos²θ .
⇒ cos²θ - 2cosθsinθ = sin²θ .
[ Adding sin²θ both side, we get ] .
⇒ cos²θ - 2cosθsinθ + sin²θ = sin²θ + sin²θ .
⇒ ( cos θ - sin θ )² = 2sin²θ .
⇒ cos θ - sin θ = √( 2sin²θ ) .
∴ cos θ - sin θ = √2sin θ ......
Hence, it is proved .
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