how do I solve these?
(Tan^2ө)(Cos^2ө)+ (cot^2ө)(sin^2ө)
The answer is supposed to be one but I don't know how to get it.
9sec^2ө - 5 tan^2ө
The answer is supposed to be 5+4sec^2ө
(Tan^2ө)(Cos^2ө)+ (cot^2ө)(sin^2ө)
sin^2 cos^2/cos^2 + (cos^2/sin^2) sin^2
sin^2 + cos^2
1
To solve these trigonometric expressions, we will use some trigonometric identities to simplify them.
Let's start with the first expression: (Tan^2ө)(Cos^2ө) + (cot^2ө)(sin^2ө).
We can start by using the identity for tangent, which states that tan^2ө = 1 - cos^2ө.
Plugging this into the expression, we get (1 - cos^2ө)(cos^2ө) + (cot^2ө)(sin^2ө).
Next, we can simplify the expression (1 - cos^2ө)(cos^2ө) by expanding it:
cos^2ө - cos^4ө + cot^2ө sin^2ө.
Now, let's use another identity, cot^2ө = 1 + tan^2ө, and substitute it into the expression:
cos^2ө - cos^4ө + (1 + tan^2ө)(sin^2ө).
Expanding further, we have:
cos^2ө - cos^4ө + sin^2ө + tan^2ө sin^2ө.
Next, we can use the Pythagorean identity sin^2ө + cos^2ө = 1 to simplify further:
(1 - cos^2ө) + sin^2ө + tan^2ө sin^2ө.
Now, combine like terms:
1 + tan^2ө sin^2ө.
Lastly, we can use another trigonometric identity tan^2ө + 1 = sec^2ө to simplify:
sec^2ө sin^2ө.
Since sin^2ө is less than or equal to 1 and sec^2ө is always greater than or equal to 1, the result of this expression is always less than or equal to sec^2ө.
Therefore, the expression (Tan^2ө)(Cos^2ө) + (cot^2ө)(sin^2ө) will always be less than or equal to sec^2ө, which is 1 when evaluated at certain angles. Hence, the answer is indeed 1.
Moving on to the second expression: 9sec^2ө - 5 tan^2ө.
We can start by using the identity tan^2ө = sec^2ө - 1.
Plugging this into the expression, we get 9sec^2ө - 5(sec^2ө - 1).
Expanding further, we have 9sec^2ө - 5sec^2ө + 5.
Simplifying, we get 4sec^2ө + 5.
Therefore, the expression 9sec^2ө - 5 tan^2ө simplifies to 4sec^2ө + 5.
9 /cos^2 - 5 sin^2 / cos^2
(9 - 5 sin^2 )/cos^2
(9 - 5 (1-cos^2)) / cos^2
(4 + 5 cos^2)/cos^2
5 + 4 / cos^2
5 + 4 sec^2