Question #162013

Trigonometric identities

Solve and prove trigonometric identities

1.) Cot(theta)sin(theta)cos(theta)=1-sin²(theta)

2.) Sin²(theta)sec²(theta)-1

3.) Tan(theta)+cot(theta)= csc(theta)sec(theta)

4.) 1+cot(theta)= csc(theta)(cos(theta) + sin(theta))

Expert's answer

"1.)\\cot(\\theta)\\sin(\\theta)\\cos(\\theta)=1-\\sin^2(\\theta)"

"\\frac{cos(\\theta)}{sin(\\theta)}(\\theta)\\sin(\\theta)\\cos(\\theta)=1-\\sin^2(\\theta)"

"\\cos^2(\\theta)= 1-\\sin^2(\\theta)"

"\\cos^2(\\theta)+\\sin^2(\\theta)=1"

"1=1\\ \\blacksquare"

"2.)\\sin^2(\\theta)\\sec\u00b2(\\theta)-1="

"\\sin^2(\\theta)\\frac{1}{cos\u00b2(\\theta)}-1="

"\\frac{1-\\cos^2(\\theta)}{\\cos^2(\\theta)}-1=\\frac{1}{\\cos^2(\\theta)}-2="

"\\sec^2{(\\theta)}-2"

"3.)\\tan(\\theta)+\\cot(\\theta)= \\csc(\\theta)\\sec(\\theta)"

"\\frac{\\sin(\\theta)}{\\cos(\\theta)}+\\frac{\\cos(\\theta)}{\\sin(\\theta)}= \\csc(\\theta)\\sec(\\theta)"

"\\frac{\\sin^2(\\theta)+sin^2(\\theta)}{\\cos(\\theta)\\sin(\\theta)}= \\csc(\\theta)\\sec(\\theta)"

"\\frac{1}{\\cos(\\theta)\\sin(\\theta)}= \\csc(\\theta)\\sec(\\theta)"

"\\frac{1}{\\cos(\\theta)}\\frac{1}{\\sin(\\theta)}= \\csc(\\theta)\\sec(\\theta)"

"\\sec(\\theta)\\csc(\\theta)=\\csc(\\theta)\\sec(\\theta)"

"\\csc(\\theta)\\sec(\\theta)=\\csc(\\theta)\\sec(\\theta)\\ \\blacksquare"

"4.)1+\\cot(\\theta)= \\csc(\\theta)(cos(\\theta) + sin(\\theta))"

"1+\\cot(\\theta)= \\frac{1}{sin(\\theta)}(cos(\\theta) + sin(\\theta))"

"1+\\cot(\\theta)= \\frac{\\cos(\\theta)}{\\sin(\\theta)}+1"

"1+\\cot(\\theta)= \\cot(\\theta)+1"

"1+\\cot(\\theta)=1+\\cot(\\theta)\\ \\blacksquare"

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