tan/1-cot +cot/1-tan =1+sec cosec
The proper symbol used for cosecant is csc.
Please use parentheses where needed.
Is <1+sec cosec> supposed to represent
(1+ sec)*csc or , 1 + (sec*csc)> ?
You should also have used parentheses around (1-cot) and (1+tan) to avoid confusion.
It may help to change tan/(1-cot)to the equivalent 1/(cot-1), and change cot/(1-tan) to 1/(tan-1), and then rewrite the sum on the left with a common denominator.
Factor the trinomial
c^2+5c+6
To solve this fraction, we'll start by simplifying both the numerator and denominator separately.
Let's begin with the numerator:
tan/1 - cot
To simplify this, we need to express both terms in terms of either sine (sin) or cosine (cos). Since tan is the ratio of sin and cos, and cot is the reciprocal of tan, we can rewrite the numerator as:
(sin/cos) / 1 - (cos/sin)
Next, let's combine the fractions:
[(sin^2 - cos^2) / (cos * sin)] / (sin - cos) / (cos * sin)
Using the difference of squares formula (a^2 - b^2 = (a + b)(a - b)), we can simplify further:
[(sin + cos)(sin - cos) / (cos * sin)] / (sin - cos) / (cos * sin)
Since we have (sin - cos) both in the numerator and the denominator, they cancel out:
[(sin + cos) / (cos * sin)] / 1 / (cos * sin)
Now, let's simplify the denominator:
1 - tan/1 + cot
Similar to the numerator, we can rewrite this as:
1 - (sin/cos) / 1 + (cos/sin)
Combining the fractions:
(1 * cos - sin) / (cos * sin) / (sin + cos) / (cos * sin)
Which simplifies to:
(cos - sin) / (cos * sin) / (sin + cos) / (cos * sin)
Now, let's simplify further by inverting the second fraction and multiplying:
(cos - sin) / (cos * sin) * (cos * sin) / (sin + cos)
The (cos * sin) in the numerator and denominator cancel out:
(cos - sin) / (sin + cos)
Now, we can observe that the numerator (cos - sin) is equal to -(sin - cos).
-(sin - cos) / (sin + cos)
To get rid of the negative sign, we can rearrange the terms:
(cos - sin) / (sin + cos) = (cos - sin) / (cos + sin)
Finally, we can observe that (cos + sin) is equal to (sin + cos):
(cos - sin) / (sin + cos) = (cos - sin) / (sin + cos) = 1
Thus, we have proven that:
tan/1 - cot + cot/1 - tan = 1 + sec cosec