Question

(1-cos^2X)/(1-sin^2X)+tgX*ctgX РЕШИТЕ

Answer 1

1-cos^x=sin^2x
1-sin^2x=cos^2x по основному тригонометрическому тождеству
tgx= sinx/cosx
ctgx=cosx/sinx
tgx*ctgx=1

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

tg^2x+1

Answer 2

To solve the expression, we'll start by manipulating the terms step by step:

Given expression: (1 - cos^2X) / (1 - sin^2X) + tgX * ctgX

Step 1: Rewrite trigonometric functions in terms of each other

• Recall that tgX is the same as tanX, and ctgX is the same as cotX: tgX = tanX ctgX = 1 / tanX = cotX

Step 2: Replace sin^2X and cos^2X with their trigonometric identities sin^2X + cos^2X = 1

Step 3: Substitute the identities into the expression: (1 - cos^2X) / (1 - sin^2X) + tgX * ctgX = (1 - cos^2X) / (1 - (1 - cos^2X)) + tanX * cotX

Step 4: Simplify the denominators: = (1 - cos^2X) / cos^2X + tanX * cotX

Step 5: Divide the terms with the same denominator: = [(1 - cos^2X) + tanX * cotX * cos^2X] / cos^2X

Step 6: Distribute tanX * cotX: = [1 - cos^2X + tanX] / cos^2X

Step 7: Use the identity tanX = sinX / cosX: = [1 - cos^2X + sinX / cosX] / cos^2X

Step 8: Get a common denominator for the numerator: = [(cosX - cos^3X + sinX) / cosX] / cos^2X

Step 9: Invert the division and multiply by the reciprocal: = (cosX / (cosX - cos^3X + sinX)) * (1 / cos^2X)

Step 10: Cancel out common factors: = 1 / (cosX - cos^3X + sinX) * (1 / cosX)

Step 11: Use the identity cos^2X + sin^2X = 1: = 1 / (cosX - (1 - cos^2X) + sinX)

Step 12: Simplify the expression: = 1 / (cosX - 1 + cos^2X + sinX) = 1 / (cos^2X + sinX - 1)

So, the simplified expression is 1 / (cos^2X + sinX - 1).