Question

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))

Answer 1

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

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

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

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

$1=1\ \blacksquare$

$2.)\sin^2(\theta)\sec²(\theta)-1=$

$\sin^2(\theta)\frac{1}{cos²(\theta)}-1=$

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

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

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

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

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

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

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

$\sec(\theta)\csc(\theta)=\csc(\theta)\sec(\theta)$

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

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

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

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

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

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