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subject: Solving Trigonometric Identities Online [print this page]

An equation is an equality which is true only for several values of the variable. An expression can be given more clear and convenient form by an identity.

Solving Trigonometric Identities Online:

Online trigonometric identities solving shows a clear idea about a tutor solves a problem in online. Some of the trigonometric identities and example problems using trigonometric identities are listed below.

Types of Trigonometric Identities:

There are several identities in trigonometry. Here we see some of trignometric identities online. They are as follows,

Reciprocal identities,

sin x = '1 / csc x' , csc x = '1 / sin x' ,

cos x = '1 / sec x' , sec x = '1 / cos x' ,

tan x = '1 / cot x' , cot x = '1 / tan x' .

Tangent and Cotangent identities,

tan x = 'sin x / cos x' ,

cot x = 'cos x / sin x' .

Pythagorean identities,

sin2x + cos2x = 1,

1 + tan2x = sec2x,

1 + cot2x = csc2x.

Co-function identities,

sin (90 x) = cos x, cos (90 x) = sin x, tan (90 x) = cot x,

csc (90 x) = sec x, sec (90 x) = csc x, cot (90 x) = tan x.

Even-Odd identities,

sin (x) = sin x, cos (x) = cos x, tan (x) = tan x,

csc (x) = csc x, sec (x) = sec x, cot (x) = cot x.

Examples of Solving Trigonometric Identities Online:

Ex 1: Solve sin30 by using cos60 = 0.5

Sol : Solving sin 30,

sin30 = sin(90 60)

= cos60 [Using Co-Function Identity]

= 0.5

Therefore, sin30 = 0.5

Ex 2: Prove that, sec2x + csc2x = sec2x csc2x using trigonometric identities.

Proof:

Solving L.H.S. ,

L.H.S. = sec2x + csc2x

= ('1 / (cos^(2)x)' ) + ('1 / (sin^(2)x)' ) [Using Reciprocal Identity]

= ' (sin^(2)x + cos^(2)x) / (sin^(2)x cos^(2)x)' [Taking L.C.M. ]

= '1 / (sin^(2)x cos^(2)x)' [Using Pythagorean Identity]

= '1 / (sin^(2)x)' '1 / (cos^(2)x)' [Separating it Into Two Terms]

= csc2x sec2x [Using Reciprocal Identity]

= sec2x csc2x

= R.H.S.

Hence, proved that sec2x + csc2x = sec2x csc2x.

Ex 3:

Prove that, sin4 a 2sin2 a cos2 a + cos4 a = cos2 (2 a), Using Trigonometric Functions calculus.

Proof:

L.H.S. = sin4 a 2 sin2 a cos2 a + cos4 a

= (sin2 a)2 2sin2 a cos2 a + (cos2 a)2

= (sin2 a cos2 a)2 [ Using ( a b )2 = a2 2ab + b2 ]

= ((1 cos2 a) cos2 a)2 [By Pythagorean identity ]

= (1 2cos2 a)2

=( (2cos2 a 1))2

=( cos 2a)2 [ Using Formula: cos 2 = 2 cos2 1 ]

=cos2 (2 a)

= R.H.S.

Hence proved that, sin4 a 2sin2 a cos2 a + cos4 a = cos2 (2 a).

by: Smith

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