Lanjutan Identitas Triginometri

Contoh 1
Buktikan : $cs{c}^{2}xsecx=secx+cotxcscx$
Jawab :
csc²x sec x = (1 + cot²x) sec x
csc²x sec x = sec x + cot²x sec x
csc²x sec x = sec x + cot x . cot x . sec x
csc²x sec x = sec x + cot x $\cdot \frac{co\text{/}sx}{sinx}\cdot \frac{1}{co\text{/}sx}$
csc²x sec x = sec x + cot x $\cdot \frac{1}{sinx}$
csc²x sec x = sec x + cot x csc x

Contoh 2
Buktikan : sec⁴t − sec²t = tan⁴t + tan²t
Jawab :
sec⁴t − sec²t = (sec²t)² − sec²t
sec⁴t − sec²t = (1 + tan²t)² − (1 + tan²t)
sec⁴t − sec²t = 1 + 2tan²t + tan⁴t − 1 − tan²t
sec⁴t − sec²t = tan⁴t + tan²t

Contoh 3
Buktikan : sin⁴x − cos⁴x = 1 − 2cos²x
Jawab :
sin⁴x − cos⁴x = (sin²x + cos²x)(sin²x − cos²x)
sin⁴x − cos⁴x = 1 . (sin²x − cos²x)
sin⁴x − cos⁴x = sin²x − cos²x
sin⁴x − cos⁴x = 1 − cos²x − cos²x
sin⁴x − cos⁴x = 1 − 2cos²x

Contoh 4
Buktikan : $(secx-tanx{)}^2=\frac{1-sinx}{1+sinx}$
Jawab :
$\begin{array}{cc}(secx-tanx{)}^2& ={\left(\frac{1}{cosx}-\frac{sinx}{cosx}\right)}^2\\ & ={\left(\frac{1-sinx}{cosx}\right)}^2\\ & =\frac{(1-sinx{)}^2}{co{s}^{2}x}\\ & =\frac{(1-sinx{)}^2}{1-si{n}^{2}x}\\ & =\frac{(1-s\text{/}inx)(1-sinx)}{(1-s\text{/}inx)(1+sinx)}\\ & =\frac{1-sinx}{1+sinx}\end{array}$

Contoh 5
Buktikan : $\frac{2-se{c}^{2}x}{se{c}^{2}x}=1-2si{n}^{2}x$
Jawab :
$\begin{array}{cc}\frac{2-se{c}^{2}x}{se{c}^{2}x}& =\frac{2}{se{c}^{2}x}-\frac{se{c}^{2}x}{se{c}^{2}x}\\ & =2co{s}^{2}x-1\\ & =2(1-si{n}^{2}x)-1\\ & =2-2si{n}^{2}x-1\\ & =1-2si{n}^{2}x\end{array}$

Contoh 6
Buktikan : $\frac{1-cosx}{sinx}=\frac{1}{cscx+cotx}$
Jawab :
$\begin{array}{cc}\frac{1-cosx}{sinx}& =\frac{1}{sinx}-\frac{cosx}{sinx}\\ & =cscx-cotx\\ & =(cscx-cotx)\cdot \frac{cscx+cotx}{cscx+cotx}\\ & =\frac{(cscx-cotx)(cscx+cotx)}{cscx+cotx}\\ & =\frac{cs{c}^{2}x-co{t}^{2}x}{cscx+cotx}\\ & =\frac{1}{cscx+cotx}\end{array}$

Contoh 7
Buktikan : $\frac{sinx}{1+cosx}+\frac{1+cosx}{sinx}=2cscx$
Jawab :
$\begin{array}{cc}\frac{sinx}{1+cosx}+\frac{1+cosx}{sinx}& =\frac{1-cosx}{1-cosx}\cdot \frac{sinx}{1+cosx}+\frac{1+cosx}{sinx}\\ & =\frac{(1-cosx)sinx}{1-co{s}^{2}x}+\frac{1+cosx}{sinx}\\ & =\frac{(1-cosx)si\text{/}nx}{si{n}^{\text{/}2}x}+\frac{1+cosx}{sinx}\\ & =\frac{1-cosx}{sinx}+\frac{1+cosx}{sinx}\\ & =\frac{2}{sinx}\\ & =2cscx\end{array}$

Contoh 8
Buktikan : $\frac{cscx-1}{cotx}=\frac{cotx}{cscx+1}$
Jawab :
$\begin{array}{cc}\frac{cscx-1}{cotx}& =\frac{cscx-1}{cotx}\cdot \frac{cscx+1}{cscx+1}\\ & =\frac{cs{c}^{2}x-1}{cotx(cscx+1)}\\ & =\frac{co{t}^{\text{/}2}x}{c\text{/}otx(cscx+1)}\\ & =\frac{cotx}{cscx+1}\end{array}$

Contoh 9
Buktikan : $\frac{co{s}^{2}x-si{n}^{2}x}{1-ta{n}^{2}x}=co{s}^{2}x$
Jawab :
$\begin{array}{cc}\frac{co{s}^{2}x-si{n}^{2}x}{1-ta{n}^{2}x}& =\frac{co{s}^{2}x-si{n}^{2}x}{1-ta{n}^{2}x}\cdot \frac{se{c}^{2}x}{se{c}^{2}x}\\ & =\frac{co{s}^{2}x\cdot se{c}^{2}x-si{n}^{2}x\cdot se{c}^{2}x}{(1-ta{n}^{2}x)se{c}^{2}x}\\ & =\frac{co{s}^{2}x\cdot \frac{1}{co{s}^{2}x}-si{n}^{2}x\cdot \frac{1}{co{s}^{2}x}}{(1-ta{n}^{2}x)se{c}^{2}x}\\ & =\frac{1-\text{/}ta{n}^{2}x}{(1-\text{/}ta{n}^{2}x)se{c}^{2}x}\\ & =\frac{1}{se{c}^{2}x}\\ & =co{s}^{2}x\end{array}$

Contoh 10
Buktikan : $\frac{co{s}^{2}x+2cosx+1}{cosx+1}=\frac{1+secx}{secx}$
Jawab :
$\begin{array}{cc}\frac{co{s}^{2}x+2cosx+1}{cosx+1}& =\frac{(cosx+1{)}^{\text{/}2}}{(cos\text{/}x+1)}\\ & =cosx+1\\ & =(cosx+1)\cdot \frac{secx}{secx}\\ & =\frac{cosxsecx+secx}{secx}\\ & =\frac{cosx\cdot \frac{1}{cosx}+secx}{secx}\\ & =\frac{1+secx}{secx}\end{array}$

Contoh 11
Buktikan : $\sqrt{\frac{1+cosx}{1-cosx}}=cscx+cotx$
Jawab :
$\begin{array}{cc}\sqrt{\frac{1+cosx}{1-cosx}}& =\sqrt{\frac{1+cosx}{1-cosx}\cdot \frac{1+cosx}{1+cosx}}\\ & =\sqrt{\frac{(1+cosx{)}^2}{1-co{s}^{2}x}}\\ & =\sqrt{\frac{(1+cosx{)}^2}{si{n}^{2}x}}\\ & =\frac{1+cosx}{sinx}\\ & =\frac{1}{sinx}+\frac{cosx}{sinx}\\ & =cscx+cotx\end{array}$

Contoh 12
Buktikan : $\frac{si{n}^{3}x+co{s}^{3}x}{sinx+cosx}=1-sinxcosx$
Jawab :
$\begin{array}{cc}\frac{si{n}^{3}x+co{s}^{3}x}{sinx+cosx}& =\frac{(sinx\text{/}+cosx)(si{n}^{2}x+co{s}^{2}x-sinxcosx)}{(sinx\text{/}+cosx)}\\ & =si{n}^{2}x+co{s}^{2}x-sinxcosx\\ & =1-sinxcosx\end{array}$