Question

Помогите пожалуйста,срочно!!!Упростить: 1) (tgt+ctgt)2 – (tgt – ctgt)2; 2) sin2tcos2t(tg2t+ctg2t +2)

Answer 1

(tgt+ctgt)²-(tgt-ctgt)²=tg²t+2tgt*ctgt+ctg²-(tg²t-2tgt*ctgt+ctg²t)=2tgt*ctgt=4*1=4

sin2tcos2t*(tgt+ctgt+2)=1+sin4t

1. tg2t+ctg2t=sin2t/cos2t+cos2t/sin2t=(sin²2t+cos²2t)/(sin2t*cos2t)=1/(sin2t*cos2t)
2. sin2t*cos2t*(1/sin2t*cos2t+2)=sin2t*cos2t*(1+2sin2t*cos2t)/(sin2t*cos2t)=1+sin4t

Answer 2

Simplifying Mathematical Expressions

To simplify the given expressions, let's start with each part separately.

1. Simplifying (tgt + ctgt)^2 - (tgt - ctgt)^2: - We can use the identity (a + b)(a - b) = a^2 - b^2 to simplify this expression.

The given expression can be simplified as follows: ``` (tgt + ctgt)^2 - (tgt - ctgt)^2 = ((tgt)^2 + 2*tgt*ctgt + (ctgt)^2) - ((tgt)^2 - 2*tgt*ctgt + (ctgt)^2) = (tgt)^2 + 2*tgt*ctgt + (ctgt)^2 - (tgt)^2 + 2*tgt*ctgt - (ctgt)^2 = 4*tgt*ctgt ```

Therefore, (tgt + ctgt)^2 - (tgt - ctgt)^2 simplifies to 4*tgt*ctgt.

2. Simplifying sin^2t * cos^2t * (tg^2t + ctg^2t + 2): - We can use trigonometric identities to simplify this expression.

The given expression can be simplified as follows: ``` sin^2t * cos^2t * (tg^2t + ctg^2t + 2) = sin^2t * cos^2t * (sin^2t/cos^2t + cos^2t/sin^2t + 2) = sin^2t * cos^2t * ((sin^4t + cos^4t)/(sin^2t*cos^2t) + 2) = sin^2t * cos^2t * ((1 - 2*sin^2t*cos^2t)/(sin^2t*cos^2t) + 2) = sin^2t * cos^2t * (1/(sin^2t*cos^2t) - 2 + 2) = 1 - 2*sin^2t*cos^2t + 2*sin^2t*cos^2t = 1 ```

Therefore, sin^2t * cos^2t * (tg^2t + ctg^2t + 2) simplifies to 1.

Final Simplified Expressions:

1. (tgt + ctgt)^2 - (tgt - ctgt)^2 simplifies to 4*tgt*ctgt. 2. sin^2t * cos^2t * (tg^2t + ctg^2t + 2) simplifies to 1.

I hope this helps! If you have any more questions or need further assistance, feel free to ask.