Question

Prove that

(59: ; 4 :<=;2)/

(59:; 2 :<=;2)

= cosec θ + cot θ​

Answer 1

Step-by-step explanation:

Prove that cos theta – sin theta + 1 / cos theta + sin theta – 1 = cosec theta + cot theta

• Given cos theta – sin theta + 1 / cos theta + sin theta – 1
• Divide numerator and denominator by sin theta we get
•      cos theta / sin theta – 1 + 1/sin theta / cos theta / sin theta + 1 – 1/sin theta
•                    cot theta – 1 + cosec theta / cot theta + 1 – cosec theta
•                    cot theta + cosec theta – (cosec^2 theta – cot^2 theta)/ cot theta + 1 – cosec theta (since we can write cosec^2 theta – cot^2 theta = 1)
•                cot theta + cosec theta – (cosec theta – cot theta ) (cosec theta + cot theta) / 1 – cosec theta + cot theta (since a^2 – b^2 = (a + b) (a – b)
•                        Taking cot theta + cosec theta as common factor we get
•          cot theta + cosec theta (1 – (cosec theta – cot theta) ) / 1 – cosec theta + cot theta
•           cot theta + cosec theta (1 – cosec theta + cot theta) ) / 1 – cosec theta + cot theta
•                                          = cot theta + cosec theta  (proved)

Reference link will be

https://brainly.in/question/2632971

https://brainly.in/question/33178940

Answer 2

Step-by-step explanation:

Prove that cos theta – sin theta + 1 / cos theta + sin theta – 1 = cosec theta + cot theta

Given cos theta – sin theta + 1 / cos theta + sin theta – 1

Divide numerator and denominator by sin theta we get

    cos theta / sin theta – 1 + 1/sin theta / cos theta / sin theta + 1 – 1/sin theta

                  cot theta – 1 + cosec theta / cot theta + 1 – cosec theta

                  cot theta + cosec theta – (cosec^2 theta – cot^2 theta)/ cot theta + 1 – cosec theta (since we can write cosec^2 theta – cot^2 theta = 1)

              cot theta + cosec theta – (cosec theta – cot theta ) (cosec theta + cot theta) / 1 – cosec theta + cot theta (since a^2 – b^2 = (a + b) (a – b)

                      Taking cot theta + cosec theta as common factor we get

        cot theta + cosec theta (1 – (cosec theta – cot theta) ) / 1 – cosec theta + cot theta

         cot theta + cosec theta (1 – cosec theta + cot theta) ) / 1 – cosec theta + cot theta

                                        = cot theta + cosec theta  (proved)

Reference link will be

brainly.in/question/2632971

brainly.in/question/33178940