Trigonometric Proof (A Level only) (OCR A Level Maths: Pure)

Given the identity

sin² cos² 

prove the following identities:

sec² tan² 

cosec² cot² 

cos  cos⁴  cos² 

sin  sin⁴  sin² 

Given that is small, and that terms involving or higher powers of  can be ignored, use an appropriate approximation to show that

cos cos² 

Show that the result in part (a) gives a percentage error of 0.583%, to 3 significant figures, when used to approximate

coscos² 

Prove the identity

Show that

sin 

Use the quotient rule to show that

Show that

sin sin  sin sin³ 

Prove the identity

Show that

sin

Show that

Show that

Given the identity

coscos  cos sin  sin 

prove the following identities:

cos cos² sin² 

cos sin² 

cos cos² 

sin  sin  sin³ 

cos  cos  cos³ 

Given that is small, and that terms involving or higher powers of  can be ignored, show that

Determine the percentage error when the result in part (a) is used to approximate

giving your answer correct to 3 significant figures.

Show that

cos cos  sin⁴  sin² 

Prove that

cot² tan²  cot  cosec .

Prove the identity

Prove the identity

cosec

Show that

tan

Given that , where is a constant, show that    .

Hence show that the value of is constant and independent of .
Find the value of this constant.

Show that

where

Consider the three triangles, all of height 1, as shown below.

By applying the area of a triangle formula  to each one, prove that,

sinsin  cos sin  cos 

Briefly explain why this only proves the result for and being acute angles.

Prove the identity

Use the small angle approximations for sine and cosine to confirm the following two limit results:

Be sure to explain why use of the small angle approximations is justified here.

Hence prove from first principles that

Prove the identity

cot  cosec³ sec⁴ cosec⁴ 

Show that

Show that

sin  sin  cos² sin³ 

Hence, or otherwise, show that

Show that

cos² sin²  sin 

Show that

tansec tan

On the axes below sketch the graphs of

  and  

With the help of your sketch, determine the exact solution(s) to the equation

What can you say about the solution(s) to the equation

Justify your answer.

Show that