Question

sec²theta+ cosec² (180 - 0) = sec²theta. cosec2theta​

Answer 1

Answer:√3

Explanation:

sin (90° + θ) = cos θ

cos (90° + θ) = - sin θ

tan (90° + θ) = - cot θ

csc (90° + θ) = sec θ

sec ( 90° + θ) = - csc θ

cot ( 90° + θ) = - tan θ

 

Using the above proved results we will prove all six trigonometrical ratios of (180° + θ).

sin (180° + θ) = sin (90° + 90° + θ)

                   = sin [90° + (90° + θ)]

                   = cos (90° + θ), [since sin (90° + θ) = cos θ]

Therefore, sin (180° + θ) = - sin θ, [since cos (90° + θ) = - sin θ]

 

cos (180° + θ) = cos (90° + 90° + θ)

                    = cos [90° + (90° + θ)]

                    = - sin (90° + θ), [since cos (90° + θ) = -sin θ]

Therefore, cos (180° + θ) = - cos θ,  [since sin (90° + θ) = cos θ]

 

tan (180° + θ) = cos (90° + 90° + θ)

                   = tan [90° + (90° + θ)]

                   = - cot (90° + θ), [since tan (90° + θ) = -cot θ]

Therefore, tan (180° + θ) = tan θ, [since cot (90° + θ) = -tan θ]

csc (180° + θ) = 1sin(180°+Θ)

                    = 1−sinΘ, [since sin (180° + θ) = -sin θ]

Therefore, csc (180° + θ) = - csc θ;

sec (180° + θ) = 1cos(180°+Θ)

                    = 1−cosΘ, [since cos (180° + θ) = - cos θ]

Therefore, sec (180° + θ) = - sec θ

and

cot (180° + θ) = 1tan(180°+Θ)

                   = 1tanΘ, [since tan (180° + θ) =  tan θ]

Therefore, cot (180° + θ) =  cot θ

Solved example:

1. Find the value of sin 225°.

Solution:

sin (225)° = sin (180 + 45)°

             = - sin 45°; since we know sin (180° + θ) = - sin θ

             = - 1√2

2. Find the value of sec 210°.

Solution:

sec (210)° = sec (180 + 30)°

             = - sec 30°; since we know sec (180° + θ) = - sec θ

             = - 1√2

3. Find the value of tan 240°.

Solution:

tan (240)° = tan (180 + 60)°

             = tan 60°; since we know tan (180° + θ) = tan θ

             = √3