Short Tricks on Trigonometric Identities by by Top Competitive Institute for Banking, SSC, Railway

Short Tricks on Trigonometric Identities

Pythagorean Identities

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

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

TRICK-1

If A + B = 90^o, 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:

Tan A = Cot B, Tan A*Tan B = 1

So, A +B = 90^o

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

Tan (2x/3) = tan 30^o = 1/√3

TRICK-2

If A – B = 90^o, Then

Sin A = Cos B
Cos A = – Sin B
Tan A = – Cot B

If A ± B = 180^o, then

Sin A = Sin B
Cos A = – Cos B

If A + B = 180^o

Then, tan A = – tan B

If A – B = 180^o

Then, tan A = tan B

For example:

Find the Value of tan 80^o + tan 100^o ?

Solution:

Since 80 + 100 = 180

Therefore, tan 80^o + tan 100^o = 1

TRICK-3

If A + B + C = 180^o, 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 20^o cos 40^o cos 60^o cos 80^o?

Solution:

We know cos θ * cos 2θ * cos 4θ = ¼ cos 3θ

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

¼ (Cos 3*20) * cos 60^o

¼ Cos² 60^o= ¼ * (½)² = 1/16

TRICK-4

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 2 cos θ – 3 sin θ = x

By using formulae a² + b² = m² + n²

4² + 3² = 2² + x²

⇒16 + 9 = 4 + x²

⇒X = √21

TRICK-5

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) = 3/2 = 2/q

Q = 4/3 or csc θ – sec θ = 4/3

TRICK-6

If a cot θ + b csc θ = m & a csc 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

y = a² sin²x + b² csc²x + c

y = a² cos²x + b² sec²x + c

y = a² tan²x + b² cot²x + c

then,

y_min = 2ab + c

y_max = not defined

For Example:

If y = 9 sin² x + 16 csc² x +4 then y_min is:

Solution:

For, y min = 2* √9 * √16 + 4

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

TRICK-7

y = a sin x + b cos x + c

y = a tan x + b cot x + c

y = a sec x + b csc x + c

then, y_min = + [√(a²+b²)] + c

y_max = – [√(a²+b²)] + c

For Example:

If y = 1/(12sin x + 5 cos x +20) then y_max 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

Important questions of Trigonometric identities

(1)Value of is

(a)

(b)

(c)

(d)None of these

Ans.(a)

is equal to

(2)If is acute and then is equal to

(a)

(b)3

(d) 4

Ans. (c)

If sum of the inversely proportional value is 2

i.e if . then

so =2

or we can put = 45°

(3)The simplified value of is

(a)-1

(b)0

(c)sec²x

(d)1

Ans. (d)

The simplified value of

is obtained by putting x=y=45°

(4) Find the value of

(a) 1

(b) -1

(d) -2

Ans. (c)

put

(5) If then is equal

(a)7/4

(b) 7/2

(d)5/4

Ans. (d)

as we know that

on solving we get sec= 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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Short Tricks on Trigonometric Identities