If ( cos ᶿ + sin ᶿ)= �ã2 cos ᶿ, show that (cos ᶿ-sin ᶿ) = �ã2sin ᶿ
no idea ,so pls help
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To prove that (cos ᶿ - sin ᶿ) is equal to √2 sin ᶿ when (cos ᶿ + sin ᶿ) is equal to √2 cos ᶿ, we can use trigonometric identities.
First, let's square both sides of the given equation:
(cos ᶿ + sin ᶿ)² = (√2 cos ᶿ)²
Expanding the squares:
cos² ᶿ + 2cos ᶿ sin ᶿ + sin² ᶿ = 2(cos² ᶿ)
Now, let's simplify the equation by using the identity sin² ᶿ = 1 - cos² ᶿ:
cos² ᶿ + 2cos ᶿ sin ᶿ + (1 - cos² ᶿ) = 2(cos² ᶿ)
Combining like terms:
2cos ᶿ sin ᶿ + 1 = 2cos² ᶿ
Rearranging the terms:
2cos ᶿ sin ᶿ = 2cos² ᶿ - 1
Now, let's focus on the left side of the equation. We know that sin 2ᶿ = 2sin ᶿ cos ᶿ, so we can rewrite the equation as:
sin 2ᶿ = 2cos² ᶿ - 1
Next, let's solve for cos 2ᶿ. We can use the identity cos 2ᶿ = 1 - 2sin² ᶿ:
cos 2ᶿ = 1 - 2sin² ᶿ
Substituting this value in the previous equation:
sin 2ᶿ = cos 2ᶿ
Since the sine and cosine functions are equal, the angles must be related by a complementary relationship. In other words, 2ᶿ and 90° - 2ᶿ are complementary angles.
Therefore, we can conclude that 2ᶿ = 90° - 2ᶿ.
Dividing both sides by 2:
ᶿ = (90° - 2ᶿ)/2
Simplifying:
ᶿ = 45° - ᶿ
Now, let's focus on the expression we want to prove: (cos ᶿ - sin ᶿ).
Substitute the value of ᶿ we found:
(cos (45° - ᶿ) - sin (45° - ᶿ))
Using the cosine and sine difference identities:
(cos 45° cos ᶿ + sin 45° sin ᶿ) - (sin 45° cos ᶿ - cos 45° sin ᶿ)
Since sin 45° and cos 45° are equal to √2/2:
((√2/2) cos ᶿ + (√2/2) sin ᶿ) - ((√2/2) cos ᶿ - (√2/2) sin ᶿ)
Combining like terms:
(√2/2)(cos ᶿ + sin ᶿ) - (√2/2)(cos ᶿ - sin ᶿ)
Remembering the given equation: (cos ᶿ + sin ᶿ) = √2 cos ᶿ:
(√2/2)(√2 cos ᶿ) - (√2/2)(cos ᶿ - sin ᶿ)
Simplifying:
√2 cos ᶿ - (√2/2)(cos ᶿ - sin ᶿ)
Finally, we can see that the expression (√2/2)(cos ᶿ - sin ᶿ) is equal to √2 sin ᶿ, which proves that (cos ᶿ - sin ᶿ) = √2 sin ᶿ when (cos ᶿ + sin ᶿ) = √2 cos ᶿ.