Is csc(θ)+sin(θ)=(2-cos^2(θ))/(sin(θ)) ?

The answer to whether csc(θ)+sin(θ)=(2-cos^2(θ))/(sin(θ)) is True

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prove csc(θ)+sin(θ)=(2-cos^2(θ))/(sin(θ))

prove

csc (θ) + sin (θ) = \frac{2 - cos ^{2} ( θ )}{sin ( θ )}

True

Solution steps

csc (θ) + sin (θ) = \frac{2 - cos ^{2} ( θ )}{sin ( θ )}

Manipulating left side

csc (θ) + sin (θ)

Express with sin, cos

csc (θ) + sin (θ)

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

= \frac{1}{sin ( θ )} + sin (θ)

Simplify

\frac{1}{sin ( θ )} + sin (θ) : \frac{1 + sin ^{2} ( θ )}{sin ( θ )}

\frac{1}{sin ( θ )} + sin (θ)

Convert element to fraction:

sin (θ) = \frac{sin ( θ ) sin ( θ )}{sin ( θ )}

= \frac{1}{sin ( θ )} + \frac{sin ( θ ) sin ( θ )}{sin ( θ )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 + sin ( θ ) sin ( θ )}{sin ( θ )}

1 + sin (θ) sin (θ) = 1 + sin^{2} (θ)

1 + sin (θ) sin (θ)

sin (θ) sin (θ) = sin^{2} (θ)

sin (θ) sin (θ)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin (θ) sin (θ) = sin^{1 + 1} (θ)

= sin^{1 + 1} (θ)

Add the numbers:

1 + 1 = 2

= sin^{2} (θ)

= 1 + sin^{2} (θ)

= \frac{1 + sin ^{2} ( θ )}{sin ( θ )}

= \frac{1 + sin ^{2} ( θ )}{sin ( θ )}

= \frac{1 + sin ^{2} ( θ )}{sin ( θ )}

Manipulating right side

\frac{2 - cos ^{2} ( θ )}{sin ( θ )}

Rewrite using trig identities

\frac{2 - cos ^{2} ( θ )}{sin ( θ )}

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= \frac{2 - ( 1 - sin ^{2} ( θ ) )}{sin ( θ )}

Expand

2 - (1 - sin^{2} (θ)) : sin^{2} (θ) + 1

2 - (1 - sin^{2} (θ))

- (1 - sin^{2} (θ)) : - 1 + sin^{2} (θ)

- (1 - sin^{2} (θ))

Distribute parentheses

= - (1) - (- sin^{2} (θ))

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 1 + sin^{2} (θ)

= 2 - 1 + sin^{2} (θ)

Subtract the numbers:

2 - 1 = 1

= sin^{2} (θ) + 1

= \frac{sin ^{2} ( θ ) + 1}{sin ( θ )}

= \frac{sin ^{2} ( θ ) + 1}{sin ( θ )}

We showed that the two sides could take the same form

\Rightarrow True

p ro v e tan^{2} (x) - sin^{2} (x) = tan^{2} (x)

p ro v e sin (x) ((1 + cot^{2} (x))) = csc (x)

p ro v e \frac{1}{tan ^{2} ( x )} = \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

p ro v e sin (3 θ) + sin (θ) = 2 sin (2 θ) cos (θ)

p ro v e - cot (x) + sec (x) csc (x) = tan (x)

Is csc(θ)+sin(θ)=(2-cos^2(θ))/(sin(θ)) ?
The answer to whether csc(θ)+sin(θ)=(2-cos^2(θ))/(sin(θ)) is True