Simplify the following expression:

Question

(cot² theta + 1) (tan theta)

Answer 1

in my work, I used Φ instead of θ, but its just a variable 

 

Trig Identities

cotΦ = cosΦ/sinΦ , tanΦ = sinΦ/cosΦ

cos²Φ + sin²Φ = 1, 2sinΦcosΦ = sinΦ =>
                                      sinΦcosΦ = 1/2 (sin(2Φ)

1/sinΦ = cscΦ 
cot²Φ = cos²Φ/sin²Φ

 

(cot²(theta) + 1) (tan theta)

=(cos²Φ/sin²Φ + 1 )(sinΦ/cosΦ)  [multiple the tanΦ through]

=cosΦ/sinΦ + sinΦ/cosΦ  [find common base, then add]

=cos²Φ + sin²Φ/ (sinΦcosΦ)  [substitute with trig. identities]

= 1/(1/2 sin(2Φ) 

= 2csc(2Φ)

 

 

Answer 2

Let    x=theta

 

 

Use the identity   1 + cot² x = csc²x

 

 

Therefore,

 

(csc²x)(tanx) =

 

(1 / sin²x)(sinx / cosx) =

 

1 / (sinxcosx)