Please help me how to do this:
1) 1 + sina*cosa / (cosa + sina)^2 =
2) sina + cosa / 1- 2sin^2a =
Are you sure the first one isn't
( 1 + 2sina cosa)/(cosa + sina)^2 ?
because then it would be
= (sin^2a + cos^2a + 2sina cosa)/(cosa + sina)^2
= (sina + cosa)^2/(cosa + sina)^2
= 1
2nd:
(sina + cosa)/(1 - 2sin^2a)= (sina + cosa)/(sin^2a + cos^2a - 2sin^2a)
= sina+cosa)/( cos^2a - sin^2a)
= (sina + cosa)/( cosa - sina)(cosa+sina) )
= 1/(cosa - sina)
oh yes, I wrote the first one wrong.
THANK YOU
Sure! Let's solve these equations step by step.
1) To simplify the expression 1 + sina*cosa / (cosa + sina)^2, we can start by simplifying the numerator and the denominator separately.
Numerator:
1 + sina*cosa
We can write sina*cosa as sin(a)*cos(a) using the trigonometric identity.
Numerator:
1 + sin(a)*cos(a)
Denominator:
(cosa + sina)^2
We can expand (cosa + sina)^2 using the formula (a + b)^2 = a^2 + 2ab + b^2.
Denominator:
cos^2(a) + 2cos(a)*sin(a) + sin^2(a)
Now, let's simplify the expression further:
cos^2(a) + 2cos(a)*sin(a) + sin^2(a) can be simplified as 1 + 2cos(a)*sin(a) using the trigonometric identity sin^2(a) + cos^2(a) = 1.
So, the denominator becomes:
1 + 2cos(a)*sin(a)
Now we can substitute the simplified numerator and denominator back into the equation:
(1 + sin(a)*cos(a)) / (1 + 2cos(a)*sin(a))
That's the simplified expression for 1 + sina*cosa / (cosa + sina)^2.
2) To simplify the expression sina + cosa / 1 - 2sin^2(a), we can start by simplifying the numerator and denominator separately.
Numerator:
sina + cosa
Denominator:
1 - 2sin^2(a)
Now let's simplify the numerator and denominator further:
We can write sin^2(a) as (1 - cos^2(a)) using the trigonometric identity sin^2(a) + cos^2(a) = 1.
Numerator:
sina + cosa
Denominator:
1 - 2(1 - cos^2(a))
Simplifying the expression further:
Numerator:
sina + cosa
Denominator:
1 - 2 + 2cos^2(a)
Denominator:
-1 + 2cos^2(a)
Now we can substitute the simplified numerator and denominator back into the equation:
(sina + cosa) / (-1 + 2cos^2(a))
That's the simplified expression for sina + cosa / 1 - 2sin^2(a).
Remember, when simplifying expressions involving trigonometric functions, it's important to apply the appropriate trigonometric identities and simplify each term before combining them.