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Popular Trigonometry >

prove csc(-x)-sin(-x)=-cos(x)cot(x)

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Solution

prove

Solution

True
Solution steps
csc(−x)−sin(−x)=−cos(x)cot(x)
Manipulating left sidecsc(−x)−sin(−x)
Use the negative angle identity: sin(−x)=−sin(x)=csc(−x)−(−sin(x))
Use the negative angle identity: csc(−x)=−csc(x)=−(−sin(x))−csc(x)
Express with sin, cos
−(−sin(x))−csc(x)
Use the basic trigonometric identity: csc(x)=sin(x)1​=−(−sin(x))−sin(x)1​
Simplify −(−sin(x))−sin(x)1​:sin(x)sin2(x)−1​
−(−sin(x))−sin(x)1​
Apply rule −(−a)=a=sin(x)−sin(x)1​
Convert element to fraction: sin(x)=sin(x)sin(x)sin(x)​=sin(x)sin(x)sin(x)​−sin(x)1​
Since the denominators are equal, combine the fractions: ca​±cb​=ca±b​=sin(x)sin(x)sin(x)−1​
sin(x)sin(x)−1=sin2(x)−1
sin(x)sin(x)−1
sin(x)sin(x)=sin2(x)
sin(x)sin(x)
Apply exponent rule: ab⋅ac=ab+csin(x)sin(x)=sin1+1(x)=sin1+1(x)
Add the numbers: 1+1=2=sin2(x)
=sin2(x)−1
=sin(x)sin2(x)−1​
=sin(x)sin2(x)−1​
=sin(x)−1+sin2(x)​
Rewrite using trig identities
sin(x)−1+sin2(x)​
Use the Pythagorean identity: 1=cos2(x)+sin2(x)1−sin2(x)=cos2(x)=sin(x)−cos2(x)​
Apply the fraction rule: b−a​=−ba​=−sin(x)cos2(x)​
=−sin(x)cos2(x)​
Rewrite using trig identities
=−cos(x)sin(x)cos(x)​
Use the basic trigonometric identity: sin(x)cos(x)​=cot(x)−cos(x)cot(x)
−cos(x)cot(x)
We showed that the two sides could take the same form⇒True

Popular Examples

cot((4pi)/3)cot(x)=1,0<= x<= 2pisin^2(x)= 1/4tan(-pi/4)sqrt(3)tan(θ)+1=0

Frequently Asked Questions (FAQ)

  • Is csc(-x)-sin(-x)=-cos(x)cot(x) ?

    The answer to whether csc(-x)-sin(-x)=-cos(x)cot(x) is True
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