I don't understand how to do these particular ones? can someone help? I get the basic steps but I always get stuck.
(sin^2ө- Tan^2ө)/ (1-sec^2ө)
(secө-Tanө)^2 (1+sinө)-1
In most cases I suggest changing everything to sines and cosines, but sometimes the variations of the
Pythagorean identities come in handy
that is, by dividing sin^2 A + cos^2 A = 1 by cos^2 A we get
tan^2 A + 1 = sec^2 A
so the denominator of the first one is -tan^2 e
and the expression is
(sin^2 e - tan^2 e)/(-tan^2 e)
= sin^ e/-tan^2 e + tan^2 e/tan^2 e
= -cos^2 e + 1
= -sin^2 e
last line should be
= + sin^2 e
for the second one, I did change to sines and cosines
and got it to reduce to
(1-sin e)/(1+sin e)^2
I don't know how much further you have to go
Of course! I can help you understand how to simplify these expressions. Let's take them one by one.
Expression 1: (sin^2ө - Tan^2ө) / (1 - sec^2ө)
To simplify this expression, we'll use trigonometric identities.
Step 1: Start by using the identity tan^2ө = sin^2ө / cos^2ө. This will allow us to express tan^2ө in terms of sin^2ө.
(sin^2ө - (sin^2ө / cos^2ө)) / (1 - (1 / cos^2ө))
Step 2: Simplify the expression by finding the least common denominator (LCD), which is cos^2ө.
((sin^2ө * cos^2ө - sin^2ө) / cos^2ө) / ((cos^2ө - 1) / cos^2ө)
Step 3: Combine like terms in the numerator.
(sin^2ө * (cos^2ө - 1)) / cos^2ө / (cos^2ө - 1) / cos^2ө
Step 4: Cancel the common factors in the numerator and denominator.
sin^2ө / 1
Therefore, the simplified expression is sin^2ө.
Expression 2: (secө - Tanө)^2 (1 + sinө) - 1
Similar to the first expression, we'll use trigonometric identities to simplify this expression.
Step 1: Start by expanding the square term.
(secө - Tanө) * (secө - Tanө) * (1 + sinө) - 1
This simplifies to:
(sec^2ө - 2secөTanө + Tan^2ө) * (1 + sinө) - 1
Step 2: Substitute sec^2ө with 1 + Tan^2ө using the identity sec^2ө = 1 + Tan^2ө.
((1 + Tan^2ө) - 2secөTanө + Tan^2ө) * (1 + sinө) - 1
Step 3: Combine like terms in the numerator.
(2Tan^2ө - 2secөTanө + 1) * (1 + sinө) - 1
Step 4: Distribute (multiply) the terms.
(2Tan^2ө - 2secөTanө + 1) + (2Tan^2өsinө - 2secөTanөsinө + sinө) - 1
Step 5: Combine like terms.
4Tan^2өsinө - 4secөTanөsinө + 2Tan^2ө - 2secөTanө + sinө - 1
And that's the simplified expression.
I hope these explanations help you understand the process! If you have further questions, feel free to ask.