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shtirl [24]
3 years ago
10

Prove it :

Mathematics
1 answer:
solmaris [256]3 years ago
5 0

Heya!

\text{First, manipulate the left side.}\\\text{Use the rule:}~a-b=\frac{(a-b)(a-b)}{a+b} = \frac{a^2-b^2}{a+b} \\\text{Use the identity:}~1-sin^2(x)=cos^2(x)\\\frac{1+cos(x)-sin(x)}{1+cos(x)+sin(x)}  = \frac{cos(x)+\frac{cos^2(x)}{1+sin(y)}}{1+cos(x)+sin(x)}

\text{Second, simplify the numerator}~cos(x)+\frac{cos^2(x)}{1+sin(x)}\\\text{Convert element to a fraction:}~cos(x)=\frac{cos(x)(1+sin(x))}{1+sin(x)}\\\text{Add:}~cos(x)=\frac{cos(x)(1+sin(x))}{1+sin(x)}+\frac{cos^2(x)}{1+sin(x)}\\\text{The denominators are equal so combine:}~\frac{cos(x)(1+sin(x))+cos^2(x)}{1+sin(x)}\\\text{Simplify:} \frac{\frac{cos(x)(sin(x)+1)+cos^2(x)}{1+sin(x)} }{1+cos(x)+sin(x)}\\ \text{Apply fraction rule:}~\frac{cos(x)(1+sin(x))+cos^2(x)}{1+sin(x)}\\\\

\text{Factor:}~\frac{cos(x)(1+sin(x)+cos(x))}{(1+sin(x))(1+cos(x)+sin(x))} \\\text{Simplify:}~\frac{cos(x)}{1+sin(x)}

\text{Third, manipulate the right side.}\\\text{Use the basic trigonometric identity:}~sec(x)=\frac{1}{cos(x)} \\\text{Use the basic trigonometric identity:}~tan(x)=\frac{sin(x)}{cos(x)} \\\text{Put the expression back together:}~\frac{1}{cos(x)}-\frac{sin(x)}{cos(x)}\\\text{Simplify:}~\frac{\frac{cos^2(x)}{1+sin(x)} }{cos(x)}

\text{Fourth, simplify.}\\\text{Apply the fraction rule:}~\frac{cos^2(x)}{(1+sin(x))cos(x)} \\\text{Cancel out the common factor:}~\frac{cos(x)}{1+sin(x)}

Therefore, the expression is TRUE.

Best of Luck!

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3 years ago
Need help please !!!
irina1246 [14]

sin^2 x + 4 sinx +3       3 + sinx

--------------------------  =  -------------------

cos^2 x                        1 - sinx


factor the numerator


(sinx +3) (sinx+1)       3 + sinx

--------------------------  =  -------------------

cos^2 x                        1 - sinx


cos^2 = 1-sin^2x

(sinx +3) (sinx+1)       3 + sinx

--------------------------  =  -------------------

1- sin^2x                       1 - sinx

factor the denominator

(sinx +3) (sinx+1)       3 + sinx

--------------------------  =  -------------------

(1-sinx ) (1+sinx)                   1 - sinx

cancel the common term (1+sinx)  and (sinx +1)

(sinx +3)                       3 + sinx

--------------------------  =  -------------------

(1-sinx )                           1 - sinx


reorder the first term

3+sinx                           3 + sinx

--------------------------  =  -------------------

(1-sinx )                           1 - sinx

3 0
3 years ago
Between the numbers 12,21 27,36 and 45 which of the numbers are visible by both 3 and 9
Westkost [7]
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12 is too short to be multiplied by 9, 21 can be multiplied by 3 but not 9 again, 3 can’t go into 45. Which leaves 27 and 36. 3 times 9 equals 27. 3 times 12 equals 36, 9 times 4 equals 36.
7 0
3 years ago
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