Verify that the trigonometric equation is an identity 1- cosx 1+ cosx4 cotx cscx

Question

Verify that the trigonometric equation is an identity 1- cosx 1+ cosx4 cotx cscx 1 + cosx 1- cos x =-4 cot x csc x Which of the following statements establishes the identity? OA. 0 B. O C. 1+cos x (csc x-19-(csc x + 1)2-4cscx 4cscx 1-cos x 1+ cosx 1-cosx (cScx+ 1Xcscx-1) cscx-1 cotx 1-cos x 1cosx 1- cosx (secx+ 1X secx -1) secx-1 tan 2x 1+cos x (sec x-1)2-(sec x + 1)2-4sec x 4sec x cosx)-4 cosx4 cos x :1-cos 2x=-sin2x 1- cosx 1+ cosx (1- cosx)-(1 1+cos x 1-cos x (1+cos x)(1-cos x) 1 cosx 1-cosx (1+sin x)(1- sinx) 1 sin x cosx Establish the identity (1-sin%) sec 26-1 Which of the following shows the key steps in establishing the identity? OA. O B. (1-sin 2?) sec 28-csc 28-cos 28- --sec 28-tan 2? (1-sin 2?) sec 26-cos 2? sec 28-- 12-sec 28-cos"9-1 (1- sin 2e) sec20- tan 29 sec9-2sec2c 1 OD. (1-sin 2?) sec 28-sec 28-sin 2? sec 28-sec 28-tan 28-1 sin®? sin 0 cos 2 ? cos 2? cos "? O C. 1-sin-g) sec 28-tan®? sec 28- - cote

Explanation / Answer

1) ( 1- cos x ) / ( 1 + cos x ) - ( 1 + cos x ) / ( 1 - cos x ) =- 4 cot x csc x

manipulating left hand side

( 1- cosx )^2 - ( 1 + cos x )^2 / ( 1 + cos x ) ( 1- cos x)

= ( - 4 cos x ) / ( 1 - cos^2 x )

= - 4 cos x / sin^2 x

= - 4 cot x csc x

option c is coreect

2) ( 1 - sin^2 theta ) sec^2 theta = 1

sec ^2 theta - sec^2 theta sin^2 theta

sec^2 theta - tan^2 theta = 1

option d is correct

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