How Do You Prove These Trigonometric Identities?

[Cutie123]

Can someone please help me with these two questions.

Th first one is prove:

1-tan^2x
________ = cos2x
1+tan^2x

& the second one is

prove:

sinx+ sinxcot^2 = secx

 

[rebedotcca]

Can someone please help me with these two questions.

Th first one is prove:

1-tan^2x
________ = cos2x
1+tan^2x

& the second one is

prove:

sinx+ sinxcot^2 = secx

1)

1-tan^2x
________ = cos2x
1+tan^2xLHS : [1 - (sin^2x/ cos^2x) ] / [ 1 + (sin^2x/cos^2x)]

= [(cos^2x -sin^2x)/cos^2x] X [cos^2x/(cos^2x + sin^2x)]

= (cos^2x - sin^2x) / (cos^2x +sin^2x ) (Rmb cos^2x + sin^2x =1 )

= cos^2x -sin^2x

= cos^2x - (1 - cos^2x)
= 2cos^2x -1 (double angle formula)
= cos2x = RHS
for question 2, did you miss out an x beside cot^ ?

 

[kerimek]

I understand the importance of proving mathematical identities in order to establish their validity and applicability in various situations. In this case, we are dealing with trigonometric identities, which are fundamental in understanding and solving problems in mathematics and other scientific fields.

To prove the first identity, we can start by using the Pythagorean identity for tangent, which states that tan^2x + 1 = sec^2x. This can be rearranged to give 1 - tan^2x = sec^2x - 1. Substituting this into the original expression, we get:

(1 - tan^2x) / (1 + tan^2x) = (sec^2x - 1) / (1 + tan^2x)

Using the double angle identity for cosine, cos2x = cos^2x - sin^2x, we can rewrite the right side of the equation as:

(cos^2x - sin^2x) / (1 + tan^2x)

Next, we can use the reciprocal identity for tangent, cotx = cosx / sinx, to rewrite the left side of the equation as:

(cos^2x / sin^2x) / (1 + cos^2x / sin^2x)

Simplifying this further, we get:

(cos^2x / sin^2x) / ((sin^2x + cos^2x) / sin^2x)

= (cos^2x / sin^2x) / (1 / sin^2x)

= cos^2x

= cos^2x - sin^2x + sin^2x

= cos^2x - sin^2x + cos^2x cot^2x

= cos^2x + sin^2x cot^2x

= secx + sinxcot^2x

Therefore, we have proven that:

(1 - tan^2x) / (1 + tan^2x) = cos2x

Similarly, for the second identity, we can start by using the Pythagorean identity for sine, sin^2x + cos^2x = 1. This can be rearranged to give sin^2x = 1 - cos^2x. Substituting this into the original expression, we get:

sinx + sinxcot^2x = secx

(sin