Important Notes & Short Tricks on Trigonometric Identities

By Sachin Awasthi|Updated : January 31st, 2021

Today we will be covering a very important topic from the Advance Maths part of the Quantitative Aptitude section that is – Important Notes & Short Tricks on Trigonometric Identities. These identities will help in most of the government competitive exams like SSC CGL, CHSL, IB ACIO, RRB NTPC etc.

Important Short Tricks on Trigonometric Identities

Pythagorean Identities

  • sin2 θ + cos2 θ = 1
  • tan2 θ + 1 = sec2 θ
  • cot2 θ + 1 = cosec2 θ

Negative of a Function

  • sin (–x) = –sin x
  • cos (–x) = cos x
  • tan (–x) = –tan x
  • cosec (–x) = –cosec x
  • sec (–x) = sec x
  • cot (–x) = –cot x

If A + B = 90o, Then

  • Sin A = Cos B
  • Sin2A + Sin2B = Cos2A + Cos2B = 1
  • Tan A = Cot B
  • Sec A = Cosec B

For example:    

If tan (x+y) tan (x-y) = 1, then find tan (2x/3)?

Solution: 

If A+B = 90°, then       

Tan A = Cot B or Tan A×Tan B = 1

So, A +B = 90o

(x+y)+(x-y) = 90o, 2x = 90o , x = 45o

Tan (2x/3) = tan 30o = 1/√3

If A - B = 90o, and (A › B) Then,

  • Sin A = Cos B
  • Cos A = - Sin B
  • Tan A = - Cot B
  • Cosec A = - Sec B
  • Sec A = Cosec B

If A ± B = 180o, then

  • Cos A = - Cos B

If A + B = 180o                   

Then,

  • Sin A = Sin B
  • tan A = - tan B

If A - B = 180o                    

Then,

  • Sin A = - Sin B
  • tan A = tan B

 If A + B + C = 180o, then

Tan A + Tan B +Tan C = Tan A * Tan B *Tan C

 

sin θ * sin 2θ * sin 4θ = ¼ sin 3θ

cos θ * cos 2θ * cos 4θ = ¼ cos 3θ

For Example: What is the value of cos 20o cos 40o cos 60o cos 80o?

Solution: We know, cos θ * cos 2θ * cos 4θ = ¼ cos 3θ

Now, (cos 20o cos 40o cos 80o ) cos 60o

¼ (Cos 3*20) * cos 60o

¼ Cos2 60o = ¼ * (½)2 = 1/16

If  a sin θ + b cos θ = m     &    a cos θ - b sin θ = n

then a2 + b2 = m2 + n2

For Example:

If 4 sin θ + 3 cos θ = 2 , then find the value of  4 cos θ - 3 sin θ:

Solution:

Let 4 cos θ - 3 sin θ = x

By using formulae a2 + b2 = m2 + n2

42 + 32 = 22 + x2

16 + 9 = 4 + x2

X = √21

If

sin θ +  cos θ = p     &     cosec θ + sec θ = q

then p – (1/p) = 2/q

For Example:

If sin θ + cos θ = √2 , then find the value of  cosec θ + sec θ:

Solution:

By using formulae:

P – (1/p) = 2/q

√2-(1/√2) = 1/√2 = 2/q

q = 2√2 or cosec θ + sec θ = 2√2

If

a cot θ + b cosec θ = m     &    a cosec θ + b cot θ = n

then b2 - a2  = m2 - n2

If

cot θ + cos θ = x     &    cot θ - cos θ = y

then x2 - y2 = 4 √xy

If

tan θ + sin θ = x     &    tan θ - sin θ = y

then x2 - y2 = 4 √xy

If

y = a2 sin2x + b2 cosec2x + c

y = a2 cos2x + b2 sec2x + c

y = a2 tan2x + b2 cot2x + c

then,

ymin = 2ab + c

ymax = not defined

For Example:                    

If y = 9 sin2 x + 16 cosec2 x + 4 then ymin is:

Solution:            

For, y min = 2* √9 * √16 + 4

= 2*3*4 + 20 = 24 + 4 = 28

If            

y = a sin x + b cos x + c

y = a tan x + b cot x + c

y = a sec x + b cosec x + c

then,     ymin = - [√(a2+b2)] + c

ymax = + [√(a2+b2)] + c

For Example:                    

If y = 1/(12sin x + 5 cos x +20) then ymax is:

Solution:            

For, y max = 1/x min

= 1/[- (√122 +52) +20] = 1/(-13+20) = 1/7

Sin2 θ, maxima value = 1, minima value = 0

Cos2 θ, maxima value = 1, minima value = 0

Here are some important questions of Trigonometric identities.

(1)Value of image005 is

(a)image002

(b)image004

(c)image003

(d)None of these

Ans.(a)

image005 is equal to

image006

 

(2)If image007is acute and image008 then image009 is equal to

(a)

(b)3

(c)  2

(d)  4

Ans.  (c)

If sum of the inversely proportional value is 2

i.e if . image013 thenimage014

image015

 

so image009=2

or we can put image007= 45°

(3)The simplified value of(Secx Secy + tanx tany)2 - (Secx tany + tanx Secy)2 is

(a)-1

(b)0

(c)sec2x

(d)1

Ans. (d)

The simplified value of (Secx Secy + tanx tany)2 - (Secx tany + tanx Secy)is obtained by putting x = y = 45°

= (√2×√2 + 1×1)2 - (√2×1 + 1×√2)2

= (2+1)2 - (√2+√2)2

=(3)2 - (2√2)2

= 9 - 8 = 1

(4) Find the value of image024

(a)  1

(b)  -1

(c)  2

(d)  -2

Ans. (c)

put image017

image026

(5) If image033 then image028 is equal

(a)7/4

(b) 7/2

(c)5/2

(d)5/4

Ans. (d)

image012 as we know thatimage034

j

on solving we get secimage007= 5/4

Note:if x+y=a

and x-y=b

then x=(a+b)/2 and y=(x-y)/2

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Posted by:

Sachin AwasthiSachin AwasthiMember since Nov 2019
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Koti Nammi

Koti NammiJul 3, 2019

Some mistakes are there bro
Pradhyumna Gudivaka
This is some what confessed
And comprehensive
shweta pal

shweta palJul 23, 2020

Thanks sir
Ravi Deepak

Ravi DeepakAug 3, 2020

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Abhinav kumar
sin theta sin60-theta sin 60+theta =1/4 sin 3theta
Vaibhav Sharma

Vaibhav SharmaSep 30, 2020

This comment is hidden because it was marked spam.

Sanskar Sharma
jo yad he wo bhai bhool jaooge
dont read these formuas .

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SSC & Railway

CGLSSC GDDFCCILCHSLCPONTPCMTSStenoGroup DOthersCoursesMock Test
tags :SSC & RailwayQuantitative AptitudeSSC CGL OverviewSSC CGL Exam DatesSSC CGL Apply OnlineSSC CGL Eligibility CriteriaSSC CGL Job Profile

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