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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Transcript

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

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.