Use trigonometric identities to transform the left side of the equation to the right side (0 < θ < π/2).

Question

1) sinθ/cosθ + cosθ/sinθ = cscθsecθ

2) (tanB + cotB)/tanB=csc^2B

Answer 1

The second one is almost the same, just turn tan into sin/cos and cot into cos/sin and it works out the same.

Answer 2

1) sinθ/cosθ + cosθ/sinθ = cscθsecθ

tanθ + cotθ = 1/(sinθ*cosθ)

1st quad; everything is=.

Answer 3

For the first one, get a common denominator.

sinθ/cosθ(sinθ/sinθ) + cosθ/sinθ(cosθ/cosθ)

The result is:

(sin²θ+cos²θ)/(sinθcosθ)

Using the Pythagorean identities, the numerator is 1

The result is:

1/(sinθcosθ)

Using the reciprocal identities, you get the right side.

cscθsecθ

For the second one:

Substitute in all sin and cos for tan and cot

The result is:

(sinB/cosB + cosB/sinB)/(sinB/cosB)

The numerator is the same as problem 1 once you get a common denominator

[(sin²B+cos²B)/(sinBcosB)]/(sinB/cosB)

Use the Pythagorean identities again:

[1/(sinBcosB)]/(sinB/cosB)

Divide the fractions by multiplying by the reciprocal

[1/(sinBcosB)] x (cosB/sinB)

The cosB cancel out leaving 2 sins in the denominator

1/(sin²B)

Your can simplify the 2 sins using the reciprocal function and that gives you the right side

csc²B