Proving Trigonometric Identities Calculator

Solved example of proving trigonometric identities

Multiplying the fraction by $-1$

Applying the trigonometric identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$

Combine $\sec\left(x\right)+\frac{-\cos\left(x\right)}{1+\sin\left(x\right)}$ in a single fraction

Multiplying polynomials $\sec\left(x\right)$ and $1+\sin\left(x\right)$

Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$

Applying the tangent identity: $\displaystyle\tan\left(\theta\right)=\frac{\sin\left(\theta\right)}{\cos\left(\theta\right)}$

Add fraction's numerators with common denominators: $\frac{1}{\cos\left(x\right)}$ and $\frac{\sin\left(x\right)}{\cos\left(x\right)}$

Combine $\frac{1+\sin\left(x\right)}{\cos\left(x\right)}-\cos\left(x\right)$ in a single fraction

When multiplying two powers that have the same base ($\cos\left(x\right)$), you can add the exponents

Apply the identity: $1-\cos\left(x\right)^2$$=\sin\left(x\right)^2$

Divide fractions $\frac{\frac{\sin\left(x\right)^2+\sin\left(x\right)}{\cos\left(x\right)}}{1+\sin\left(x\right)}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$

Multiplying polynomials $\cos\left(x\right)$ and $1+\sin\left(x\right)$

Factoring by $\sin\left(x\right)$

Factoring by $\cos\left(x\right)$

Simplify the fraction $\frac{\sin\left(x\right)\left(\sin\left(x\right)+1\right)}{\cos\left(x\right)\left(1+\sin\left(x\right)\right)}$ by $\sin\left(x\right)+1$

Apply the identity: $\frac{\sin\left(x\right)}{\cos\left(x\right)}$$=\tan\left(x\right)$

Since both sides of the equality are equal, we have proven the identity