Examples of trig functions

Trig functions generally define the function of angles. To relate the angles of triangle to its length of sides is the common use of trig functions. Trig functions are generally applied in modeling periodic phenomena and study of triangles. Sine, cosine, tangent, cosecant, secant, and cotangent are the most memorable trig functions. Trig functions refer to tabulating the trig functions and trig identities in order. In  this article we shall discuss about examples involved in trig functions.

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Examples of Trig Functions:

Let us see some of the examples problems for trig function.

Examples 1: Prove that, sin^4 x − 2sin^2 x cos^2 x + cos^4 x =  cos^2 (2x).

Proof: L.H.S. = sin^4 x − 2sin^2 x cos^2 x + cos^4 x,

=> (sin^2 x)^2 − 2sin^2 x cos^2 x + (cos^2 x)2,

We know that, (a + b)^2 = a^2 − 2ab + b^2, similarly,

=> (sin^2 x − cos^2 x)^2,

=> (sin^2 x −( 1 − sin^2 x))^2,

=> (sin^2 x − 1 + sin^2 x)^2,

=> (2sin^2 x − 1)^2,

=> (−1 (1 − 2sin^2 x))^2,

=> (1 − 2sin^2 x)^2,

=> (cos(2x))^2,

=> cos^2(2x).

=> R.H.S.

Hence Proved that, sin^4 x − 2sin^2 x cos^2 x + cos^4 x = cos^2(2x).


Examples 2: Prove that, tan^2 x + 1 = sec^2 x.

Proof: We know that, sin^2 x + cos^2 x = 1

Dividing the above equation by cos^2 x, we get,

=> `(sin^2x + cos^2x) / (cos^2x) = 1 / (cos^2x)` ,

=> `(sin^2x) / (cos^2x) + (cos^2x) / (cos^2x) = 1 / (cos^2x)` ,

By using the trigonometric functions; `sinx/cosx` = tan x, and `1/cosx` = secx, we get,

=> tan^2 x + 1 = sec^2 x

Hence proved that, tan^2 x + 1 = sec^2 x.

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Examples 3: Prove that, `(tan A + cot B)/(cotA + tanB) = tan A /tanB` .

Proof: L.H.S. = `(tan A + cot B)/(cotA + tanB)` ,

We know that, cot x = `1/tanx` , we get,

=> `((tan A + 1/tanB))/((1/tanA + tanB))` ,

By taking L.C.M. We get,

=> `(((tanAxxtanB + 1)/tanB))/(((1 + tanAxxTanB)/tanA))` ,

By taking reciprocal, we get,

=>  `((tanAxxtanB + 1)/tanB)xx(tanA/(1 + tanAxxTanB))` ,

By simplification, we get,

=> `tan A /tanB` ,

=> R.H.S.

Hence proved that, `(tan A + cot B)/(cotA + tanB) = tan A /tanB` .

Practice Problems of Trigonometric Functions:

Problem 1: Prove that, `(1 + tan x)/(1+cotx) = (sin x + tanx)/(1 + cosx)`.

Problem 2: Prove that, `cosx /(1+sinx)` = sec x − tan x.

Problem 3: Prove that, `(1+cosx)/(1-cosx)` = (csc x + cot x)2.