sin (–x) = – sin x

cos (–x) = cos x

 

Note: Sometimes, these identities are called opposite angle identities

 

Since

  cos (–x) = cos x

i.e. value of cos remains same after –x,

it is called even function

 

Since

  sin (–x) = – sin x

i.e. value of sin becomes negative after –x,

 it is called odd function

 

Thus, it is called odd-even identities also

 

We need to remember only these two,

for tan, cot, cosec, sec we can find using these two.

Let’s see

 

tan

Now, since tan = sin/cos

tan (–x) = sin (–x)/cos (–x)

       = – sin x/cos x

       = – tan x

 

∴ tan (–x) = – tan x

 

sec

Now, since sec = 1/cos

sec (–x) = 1/cos (–x)

  = 1/cos x

  = sec x

 

∴ sec (–x) = sec x

 

cosec

Now, since cosec = 1/sin

cosec (–x) = 1/sin (–x)

       = 1/(– sin x)

       = – 1/sin x

       = – cosec x 

 

∴ cosec (–x) = – cosec x

 

cot

Now, since cot = 1/tan

cot (–x) = 1/tan (–x)

  = 1/(– tan x)

  = – 1/tan x

  = – cot x

 

∴ cot (–x) = – cot x

 

  1. Chapter 3 Class 11 Trigonometric Functions
  2. Concept wise
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sin (–x) = – sin x cos (–x) = cos x Note: Sometimes, these identities are called opposite angle identities Since cos (–x) = cos x i.e. value of cos remains same after –x, it is called even function Since sin (–x) = – sin x i.e. value of sin becomes negative after –x, it is called odd function Thus, it is called odd-even identities also We need to remember only these two, for tan, cot, cosec, sec we can find using these two. Let’s see tan Now, since tan = sin/cos tan (–x) = sin (–x)/cos (–x) = – sin x/cos x = – tan x ∴ tan (–x) = – tan x sec Now, since sec = 1/cos sec (–x) = 1/cos (–x) = 1/cos x = sec x ∴ sec (–x) = sec x cosec Now, since cosec = 1/sin cosec (–x) = 1/sin (–x) = 1/(– sin x) = – 1/sin x = – cosec x ∴ cosec (–x) = – cosec x cot Now, since cot = 1/tan cot (–x) = 1/tan (–x) = 1/(– tan x) = – 1/tan x = – cot x ∴ cot (–x) = – cot x

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.