I don't understand this problem:
(Tanө + cos ө)/ (sec ө + cot ө)
so I start off like this:
={(sinө / cos ө)+cosө}{cos ө + (sinө/cosө)}
=[(sin ө +cos^2ө) (cos^2ө +sin ө)]/ cos ө
but what comes next?
my second line would have been
(sin e/cos e + cos e)/(1/cos e + cos e/sin e)
= [(sin e + cos^2 e)/cos e]/[(sin e + cos^2 e)/(sin e cos e)]
= sin e
in my last step I inverted and multiplied so the sin e + cos^2 e canceled and the cos e canceled
To simplify the expression further, you can expand the numerator and simplify the expression.
Start by expanding the numerator:
(sin ө + cos^2 ө) (cos^2 ө + sin ө)
= sin ө cos^2 ө + sin ө sin ө + cos^2 ө cos^2 ө + cos^2 ө sin ө
= sin ө cos^2 ө + sin^2 ө + cos^4 ө + cos^2 ө sin ө
Now, combine like terms:
= sin^2 ө + sin ө cos^2 ө + cos^2 ө sin ө + cos^4 ө
Next, simplify the expression by factoring out sin ө:
= sin^2 ө + sin ө (cos^2 ө + cos^2 ө) + cos^4 ө
= sin^2 ө + 2sin ө cos^2 ө + cos^4 ө
Now, you need to simplify the denominator:
cos ө = 1 / sec ө
cot ө = 1 / tan ө
Substituting these into the denominator:
(sec ө + cot ө) = (1 / cos ө) + (1 / tan ө)
= (1 / cos ө) + (1 / (sin ө / cos ө))
= (1 / cos ө) + (cos ө / sin ө)
Next, find a common denominator for the two terms in the denominator:
= (1 / cos ө) + (cos ө * cos ө) / (sin ө * cos ө)
= (1 / cos ө) + (cos^2 ө) / (sin ө * cos ө)
Simplifying further:
= (1 / cos ө) + (cos^2 ө) / sin ө cos ө
= (1 / cos ө) + (cos^2 ө / sin ө cos ө)
Now, you can simplify the expression by multiplying the numerator and denominator of the fraction by the reciprocal of cos ө:
= (1 / cos ө) + (cos^2 ө / sin ө cos ө) * (1 / cos ө)
= (1 / cos ө) + (cos^2 ө / sin ө)
Finally, you can substitute the simplified numerator and denominator back into the original expression:
= (sin^2 ө + 2sin ө cos^2 ө + cos^4 ө) / ((1 / cos ө) + (cos^2 ө / sin ө))
This is the simplified form of the expression.